From acb714f5c7080d197fadc4e1a8841b85547315f6 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Sun, 5 May 2024 14:58:06 +0000 Subject: [PATCH] build based on 24b1ab0 --- stable | 2 +- v0.3 | 2 +- v0.3.2/.documenter-siteinfo.json | 1 + v0.3.2/analysis_callback.png | Bin 0 -> 20120 bytes v0.3.2/analysis_callback_relaxation.png | Bin 0 -> 29345 bytes v0.3.2/assets/documenter.js | 1050 +++++++++++++++++++++ v0.3.2/assets/themes/documenter-dark.css | 7 + v0.3.2/assets/themes/documenter-light.css | 9 + v0.3.2/assets/themeswap.js | 84 ++ v0.3.2/assets/warner.js | 52 + v0.3.2/bathymetry.png | Bin 0 -> 31582 bytes v0.3.2/development/index.html | 7 + v0.3.2/index.html | 13 + v0.3.2/license/index.html | 2 + v0.3.2/objects.inv | Bin 0 -> 1682 bytes v0.3.2/overview/index.html | 190 ++++ v0.3.2/ref-trixibase/index.html | 10 + v0.3.2/ref/index.html | 24 + v0.3.2/search_index.js | 3 + v0.3.2/shoaling_solution.gif | Bin 0 -> 555785 bytes v0.3.2/shoaling_solution.png | Bin 0 -> 32047 bytes 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.removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` +
+ + Ctrl + + / to search + + esc to close +
+ `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/v0.3.2/assets/themes/documenter-dark.css b/v0.3.2/assets/themes/documenter-dark.css new file mode 100644 index 00000000..1d717015 --- /dev/null +++ b/v0.3.2/assets/themes/documenter-dark.css @@ -0,0 +1,7 @@ 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.pagination-next:focus,html.theme--documenter-dark .pagination-link:focus,html.theme--documenter-dark .pagination-ellipsis:focus,html.theme--documenter-dark .file-cta:focus,html.theme--documenter-dark .file-name:focus,html.theme--documenter-dark .select select:focus,html.theme--documenter-dark .textarea:focus,html.theme--documenter-dark .input:focus,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:focus,html.theme--documenter-dark .button:focus,html.theme--documenter-dark .is-focused.pagination-previous,html.theme--documenter-dark .is-focused.pagination-next,html.theme--documenter-dark .is-focused.pagination-link,html.theme--documenter-dark .is-focused.pagination-ellipsis,html.theme--documenter-dark .is-focused.file-cta,html.theme--documenter-dark .is-focused.file-name,html.theme--documenter-dark .select select.is-focused,html.theme--documenter-dark .is-focused.textarea,html.theme--documenter-dark .is-focused.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--documenter-dark .is-focused.button,html.theme--documenter-dark .pagination-previous:active,html.theme--documenter-dark .pagination-next:active,html.theme--documenter-dark .pagination-link:active,html.theme--documenter-dark .pagination-ellipsis:active,html.theme--documenter-dark .file-cta:active,html.theme--documenter-dark .file-name:active,html.theme--documenter-dark .select select:active,html.theme--documenter-dark .textarea:active,html.theme--documenter-dark .input:active,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:active,html.theme--documenter-dark .button:active,html.theme--documenter-dark .is-active.pagination-previous,html.theme--documenter-dark .is-active.pagination-next,html.theme--documenter-dark .is-active.pagination-link,html.theme--documenter-dark .is-active.pagination-ellipsis,html.theme--documenter-dark .is-active.file-cta,html.theme--documenter-dark .is-active.file-name,html.theme--documenter-dark .select select.is-active,html.theme--documenter-dark .is-active.textarea,html.theme--documenter-dark .is-active.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-active,html.theme--documenter-dark .is-active.button{outline:none}html.theme--documenter-dark .pagination-previous[disabled],html.theme--documenter-dark .pagination-next[disabled],html.theme--documenter-dark .pagination-link[disabled],html.theme--documenter-dark .pagination-ellipsis[disabled],html.theme--documenter-dark .file-cta[disabled],html.theme--documenter-dark .file-name[disabled],html.theme--documenter-dark .select select[disabled],html.theme--documenter-dark .textarea[disabled],html.theme--documenter-dark .input[disabled],html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input[disabled],html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark fieldset[disabled] .pagination-previous,fieldset[disabled] html.theme--documenter-dark .pagination-next,html.theme--documenter-dark fieldset[disabled] .pagination-next,fieldset[disabled] html.theme--documenter-dark .pagination-link,html.theme--documenter-dark fieldset[disabled] .pagination-link,fieldset[disabled] html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark fieldset[disabled] .pagination-ellipsis,fieldset[disabled] html.theme--documenter-dark .file-cta,html.theme--documenter-dark fieldset[disabled] .file-cta,fieldset[disabled] html.theme--documenter-dark .file-name,html.theme--documenter-dark fieldset[disabled] .file-name,fieldset[disabled] html.theme--documenter-dark .select select,fieldset[disabled] html.theme--documenter-dark .textarea,fieldset[disabled] html.theme--documenter-dark .input,fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar 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.button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter 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#documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--documenter-dark .image.is-5by4 .has-ratio,html.theme--documenter-dark #documenter 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img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4 img,html.theme--documenter-dark .image.is-3by4 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4 .has-ratio,html.theme--documenter-dark .image.is-2by3 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3 img,html.theme--documenter-dark .image.is-2by3 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3 .has-ratio,html.theme--documenter-dark .image.is-3by5 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--documenter-dark .image.is-3by5 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--documenter-dark .image.is-9by16 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--documenter-dark .image.is-9by16 .has-ratio,html.theme--documenter-dark #documenter 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.docs-logo>img.is-5by4{padding-top:80%}html.theme--documenter-dark .image.is-4by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by3{padding-top:75%}html.theme--documenter-dark .image.is-3by2,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by2{padding-top:66.6666%}html.theme--documenter-dark .image.is-5by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by3{padding-top:60%}html.theme--documenter-dark .image.is-16by9,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-16by9{padding-top:56.25%}html.theme--documenter-dark .image.is-2by1,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by1{padding-top:50%}html.theme--documenter-dark .image.is-3by1,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--documenter-dark .image.is-4by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--documenter-dark .image.is-3by4,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4{padding-top:133.3333%}html.theme--documenter-dark .image.is-2by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3{padding-top:150%}html.theme--documenter-dark .image.is-3by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5{padding-top:166.6666%}html.theme--documenter-dark .image.is-9by16,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-9by16{padding-top:177.7777%}html.theme--documenter-dark .image.is-1by2,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by2{padding-top:200%}html.theme--documenter-dark .image.is-1by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by3{padding-top:300%}html.theme--documenter-dark .image.is-16x16,html.theme--documenter-dark #documenter .docs-sidebar 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+ Maintainer: @highlightjs/core-team + Website: https://highlightjs.org/ + License: see project LICENSE + Touched: 2021 +*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/v0.3.2/assets/themeswap.js b/v0.3.2/assets/themeswap.js new file mode 100644 index 00000000..9f5eebe6 --- /dev/null +++ b/v0.3.2/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/v0.3.2/assets/warner.js b/v0.3.2/assets/warner.js new file mode 100644 index 00000000..3f6f5d00 --- /dev/null +++ b/v0.3.2/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; + + document.getElementsByTagName("head")[0].appendChild(meta); + } + + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
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file mode 100644 index 00000000..e1d296eb --- /dev/null +++ b/v0.3.2/development/index.html @@ -0,0 +1,7 @@ + +Development · DispersiveShallowWater.jl
GitHub

Development

If you have any suggestions or ideas for improvements or new features, we are pleased to accept and discuss issues or if you are willing to contribute, feel free to open a pull request, even if it is only fixing a typo or improving the docs.

Changing DispersiveShallowWater.jl and running it locally

If you plan to edit DispersiveShallowWater.jl, you first need to clone a local copy of the repository, which can be done by using git. It is recommended that you create a project, e.g. call it run, inside the repository, where you can add packages that you use during executing and testing DispersiveShallowWater.jl, but are not needed by DispersiveShallowWater.jl. This way you can keep the Project.toml of the main repository clean. To do so, you can execute the following lines in a terminal:

git clone https://github.com/JoshuaLampert/DispersiveShallowWater.jl.git
+cd DispersiveShallowWater
+mkdir run
+cd run
+julia --project=. -e 'using Pkg; Pkg.develop(PackageSpec(path=".."))' # Install local DispersiveShallowWater.jl clone
+julia --project=. -e 'using Pkg; Pkg.add(["OrdinaryDiffEq", "Plots", "SummationByPartsOperators"])' # Install additional packages

If you use other packages for executing DispersiveShallowWater.jl, you can add them to the project in the run directory in an analogous way as above. To use the Julia project within run, be sure to start the Julia REPL by

julia --project=.

if already inside the the run directory or julia --project=run if in the main directory of the repo.

Preview of the documentation

If you want to build the documentation locally, you can run

julia --project=docs -e 'using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()'

once from the DispersiveShallowWater.jl main directory to tell Documenter.jl to build the documentation of your local clone. To build the documentation, run

julia --project=docs --color=yes docs/make.jl

The resulting .html files can then be found in docs/build/ and you can look at them by opening them in a browser.

diff --git a/v0.3.2/index.html b/v0.3.2/index.html new file mode 100644 index 00000000..fb3deebe --- /dev/null +++ b/v0.3.2/index.html @@ -0,0 +1,13 @@ + +Home · DispersiveShallowWater.jl
GitHub

DispersiveShallowWater.jl

Docs-stable Docs-dev Build Status codecov Coveralls Aqua QA License: MIT DOI

DispersiveShallowWater.jl is a Julia package that implements structure-preserving numerical methods for dispersive shallow water models. To date, it provides provably conservative, entropy-conserving and well-balanced numerical schemes for two dispersive shallow water models:

The semidiscretizations are based on summation by parts (SBP) operators, which are implemented in SummationByPartsOperators.jl. In order to obtain fully discrete schemes, the time integration methods from OrdinaryDiffEq.jl are used to solve the resulting ordinary differential equations. Fully discrete entropy-conservative methods can be obtained by using the relaxation method provided by DispersiveShallowWater.jl.

Installation

If you have not yet installed Julia, then you first need to download Julia. Please follow the instructions for your operating system. DispersiveShallowWater.jl works with Julia v1.9 and newer. DispersiveShallowWater.jl is a registered Julia package. Therefore, you can install it by executing the following commands from the Julia REPL

julia> using Pkg
+
+julia> Pkg.add(["DispersiveShallowWater", "OrdinaryDiffEq", "Plots"])

In addition, this installs the packages OrdinaryDiffEq.jl used for time-integration and Plots.jl to visualize the results. If you want to use other SBP operators than the default operators that DispersiveShallowWater.jl uses, then you also need SummationByPartsOperators.jl, which can be installed running

julia> Pkg.add("SummationByPartsOperators")

Usage

In the Julia REPL, first load the package DispersiveShallowWater.jl

julia> using DispersiveShallowWater

You can run a basic simulation that solves the BBM-BBM equations by executing

julia> include(default_example());

The result can be visualized by using the package Plots.jl

julia> using Plots
+julia> plot(semi => sol)

The command plot expects a Pair consisting of a Semidiscretization and an ODESolution. The visualization can also be customized, see the documentation for more details. Other examples can be found in the subdirectory examples/. A list of all examples is returned by running get_examples(). You can pass the filename of one of the examples or your own simulation file to include in order to run it, e.g., include(joinpath(examples_dir(), "svaerd_kalisch_1d", "svaerd_kalisch_1d_dingemans_relaxation.jl")).

Referencing

You can directly refer to DispersiveShallowWater.jl as

@misc{lampert2023dispersive,
+  title={{D}ispersive{S}hallow{W}ater.jl: {S}tructure-preserving numerical
+         methods for dispersive shallow water models},
+  author={Lampert, Joshua},
+  year={2023},
+  month={10},
+  howpublished={\url{https://github.com/JoshuaLampert/DispersiveShallowWater.jl}},
+  doi={10.5281/zenodo.10034636}
+}

Authors

The package is developed and maintained by Joshua Lampert (University of Hamburg). Some parts of this repository are based on parts of Dispersive-wave-schemes-notebooks. A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations by Hendrik Ranocha, Dimitrios Mitsotakis and David Ketcheson. The code structure is inspired by Trixi.jl.

License and contributing

DispersiveShallowWater.jl is published under the MIT license (see License). We are pleased to accept contributions from everyone, preferably in the form of a PR.

diff --git a/v0.3.2/license/index.html b/v0.3.2/license/index.html new file mode 100644 index 00000000..0d554967 --- /dev/null +++ b/v0.3.2/license/index.html @@ -0,0 +1,2 @@ + +License · DispersiveShallowWater.jl
GitHub

License

MIT License

Copyright (c) 2023-present Joshua Lampert <joshua.lampert@uni-hamburg.de> and contributors

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

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GitHub

Running a simulation

Introduction

In this tutorial we describe how to numerically solve the BBM-BBM (Benjamin-Bona-Mahony) equations with variable bottom topography in one dimension, which has been proposed in [IsrawiKalischKatsaounisMitsotakis2021] for two spatial dimensions. The equations describe a dispersive shallow water model, i.e. they extend the well-known shallow water equations in the sense that dispersion is modeled. The shallow water equations are a system of first order hyperbolic partial differential equations that can be written in the form of a balance law. In contrast, the BBM-BBM equations additionally include third-order mixed derivatives. In primitive variables $q = (\eta, v)$ they can be written as:

\[\begin{aligned} + \eta_t + ((\eta + D)v)_x - \frac{1}{6}(D^2\eta_{xt})_x &= 0,\\ + v_t + g\eta_x + \left(\frac{1}{2}v^2\right)_x - \frac{1}{6}(D^2v_t)_{xx} &= 0. +\end{aligned}\]

Here, $\eta = h + b$ describes the total water height, $h$ the water height above the bottom topography (bathymetry), $b = \eta_0 - D$ the bathymetry and $v$ the velocity in horizontal direction. Here, $\eta_0$ is a reference water height also called still water height. In the case of the BBM-BBM equations, $\eta_0$ is usually taken to be 0. The gravitational acceleration is denoted as $g$. A sketch of the water height and the bathymetry can be found below.

water height and bathymetry

In order to conduct a numerical simulation with DispersiveShallowWater.jl, we perform the following steps.

First, we load the necessary libraries:

using DispersiveShallowWater, OrdinaryDiffEq

Define physical setup

As a first step of a numerical simulation, we define the physical setup we want to solve. This includes the set of equations, potentially including physical parameters, initial and boundary conditions as well as the domain. In the following example, the initial condition describes a travelling wave that moves towards a beach, which is modeled by a linearly increasing bathymetry.

equations = BBMBBMVariableEquations1D(gravity_constant = 9.81)
+
+function initial_condition_shoaling(x, t, equations::BBMBBMVariableEquations1D, mesh)
+    A = 0.07 # amplitude of wave
+    x0 = -30 # initial center
+    eta = A * exp(-0.1*(x - x0)^2)
+    v = 0
+    D = x <= 0.0 ? 0.7 : 0.7 - 1/50 * x
+    return SVector(eta, v, D)
+end
+
+initial_condition = initial_condition_shoaling
+boundary_conditions = boundary_condition_periodic
+
+coordinates_min = -130.0
+coordinates_max = 20.0
+N = 512
+mesh = Mesh1D(coordinates_min, coordinates_max, N + 1)
Mesh1D{Float64} 
+    xmin: -130.0
+    xmax: 20.0
+    N: 513

The first line specifies that we want to solve the BBM-BBM equations with variable bathymetry using a gravitational acceleration $g = 9.81$. Afterwards, we define the initial condition, which is described as a function with the spatial variable x, the time t, the equations and a mesh as parameters. If an analytical solution is available, the time variable t can be used, and the initial condition can serve as an analytical solution to be compared with the numerical solution. Otherwise, you can just keep the time variable unused. An initial condition in DispersiveShallowWater.jl is supposed to return an SVector holding the values for each of the unknown variables. Since the bathymetry is treated as a variable (with time derivative 0) for convenience, we need to provide the value for the primitive variables eta and v as well as for D.

Next, we choose periodic boundary conditions since, up to now, DispersiveShallowWater.jl only provides support for this type of boundary conditions. Lastly, we define the physical domain as the interval from -130 to 20 and we choose 512 intermediate nodes. The mesh is homogeneous, i.e. the distance between each two nodes is constant. We choose the left boundary very far to the left in order to avoid an interaction of the left- and right-travelling waves.

Define numerical solver

In the next step, we build a Semidiscretization that bundles all ingredients for the spatial discretization of the model. Especially, we need to define a Solver. The simplest way to define a solver is to call the constructor by providing the mesh and a desired order of accuracy. In the following example, we use an accuracy order of 4. The default constructor simply creates periodic first- and second-derivative central finite difference summation by parts operators of the provided order of accuracy. How to use other summation by parts operators, is described in the section on how to customize the solver.

solver = Solver(mesh, 4)
+
+semi = Semidiscretization(mesh, equations, initial_condition, solver, boundary_conditions = boundary_conditions)
Semidiscretization
+    #spatial dimensions: 1
+    mesh: Mesh1D{Float64} with length 513
+    equations: BBMBBMVariableEquations1D
+    initial condition: initial_condition_shoaling
+    boundary condition: boundary_condition_periodic
+    source terms: nothing

Finally, we put the mesh, the equations, the initial_condition, the solver and the boundary_conditions together in a semidiscretization semi.

Solve system of ordinary differential equations

Once we have obtained a semidiscretization, we can solve the resulting system of ordinary differential equations. To do so, we specify the time interval that we want to simulate and obtain an ODEProblem from the SciML ecosystem for ordinary differential equations by calling semidiscretize on the semidiscretization and the time span. Additionally, we can analyze the numerical solution using an AnalysisCallback. The analysis includes computing the $L^2$ error and $L^\infty$ error of the different solution's variables compared to the initial condition (or, if available, at the same time analytical solution). Additional errors can be passed by the keyword argument extra_analysis_errors. Additional integral quantities that should be analyzed can be passed by keyword argument extra_analysis_integrals. In this example we pass the conservation_error, which computes the temporal change of the total amount (i.e. integral) of the different variables over time. In addition, the integrals of the total waterheight $\eta$ waterheight_total, the velocity and the entropy are computed and saved for each time step. The total waterheight and the total velocity are linear invariants of the BBM-BBM equations, i.e. they do not change over time. The total entropy

\[\mathcal E(t; \eta, v) = \frac{1}{2}\int_\Omega g\eta^2 + (\eta + D)v^2\textrm{d}x\]

is a nonlinear invariant and should be constant over time as well. During the simulation, the AnalysisCallback will print the results to the terminal.

Finally, the ode can be solved using the interface from OrdinaryDiffEq.jl. This means, we can specify a time-stepping scheme we want to use the tolerances for the adaptive time-stepping and the time values, where the solution values should be saved. In this case, we use the adaptive explicit Runge-Kutta method Tsit5 by Tsitouras of order 5(4). Here, we save the solution at 100 equidistant points.

tspan = (0.0, 25.0)
+ode = semidiscretize(semi, tspan)
+analysis_callback = AnalysisCallback(semi; interval = 10,
+                                     extra_analysis_errors = (:conservation_error,),
+                                     extra_analysis_integrals = (waterheight_total,
+                                                                 velocity, entropy),
+                                     io = devnull)
+callbacks = CallbackSet(analysis_callback)
+
+saveat = range(tspan..., length = 100)
+sol = solve(ode, Tsit5(), abstol = 1e-7, reltol = 1e-7,
+            save_everystep = false, callback = callbacks, saveat = saveat)
retcode: Success
+Interpolation: 1st order linear
+t: 100-element Vector{Float64}:
+  0.0
+  0.25252525252525254
+  0.5050505050505051
+  0.7575757575757576
+  1.0101010101010102
+  1.2626262626262625
+  1.5151515151515151
+  1.7676767676767677
+  2.0202020202020203
+  2.272727272727273
+  ⋮
+ 22.97979797979798
+ 23.232323232323232
+ 23.484848484848484
+ 23.737373737373737
+ 23.98989898989899
+ 24.242424242424242
+ 24.494949494949495
+ 24.747474747474747
+ 25.0
+u: 100-element Vector{Vector{Float64}}:
+ [0.0, 0.0, 0.7, 0.0, 0.0, 0.7, 0.0, 0.0, 0.7, 0.0  …  0.32339181286549706, 1.1159911034969113e-106, 0.0, 0.31754385964912274, 6.2568988186290324e-108, 0.0, 0.31169590643274847, 3.4485093610035348e-109, 0.0, 0.30584795321637426]
+ [-3.3298063320105386e-56, 8.240555184525373e-57, 0.7, 3.4052487839602066e-56, 3.8721596046450877e-59, 0.7, -1.2823114010005487e-56, 4.5746067309011375e-57, 0.7, 9.837872827117964e-57  …  0.32339181286549706, 9.300749704670396e-55, -7.275157739846947e-55, 0.31754385964912274, -3.348584246232203e-55, 2.8353792349816592e-55, 0.31169590643274847, 9.261972052395447e-56, -9.944024391145692e-56, 0.30584795321637426]
+ [-4.854627270617779e-53, 1.809712375822027e-53, 0.7, 4.842444655430691e-53, -1.4591991690482042e-55, 0.7, -1.8543823964202563e-53, 9.787480542672555e-54, 0.7, 1.4192813322404536e-53  …  0.32339181286549706, 1.2619264297387037e-51, -1.4764905451287523e-51, 0.31754385964912274, -4.641396873108809e-52, 5.871242596528972e-52, 0.31169590643274847, 1.320870503616519e-52, -2.0981320674476158e-52, 0.30584795321637426]
+ [-1.161236430780927e-50, 5.352584402349824e-51, 0.7, 1.132047190334612e-50, -1.090865546552179e-52, 0.7, -4.404063115451628e-51, 2.8257688001594547e-51, 0.7, 3.363556071973813e-51  …  0.32339181286549706, 2.8189227454109665e-49, -4.048320798927971e-49, 0.31754385964912274, -1.058192942359117e-49, 1.6416176052934605e-49, 0.31169590643274847, 3.0941636353866156e-50, -5.973632917118691e-50, 0.30584795321637426]
+ [-1.111984161777274e-48, 5.886881206722659e-49, 0.7, 1.0619527001701963e-48, -1.873587324635778e-50, 0.7, -4.191514844282648e-49, 3.04164693232564e-49, 0.7, 3.1953473695033716e-49  …  0.32339181286549706, 2.5336468214045444e-47, -4.1512198119827403e-47, 0.31754385964912274, -9.693390250020113e-48, 1.7143507420784363e-47, 0.31169590643274847, 2.906329540544282e-48, -6.344231008069559e-48, 0.30584795321637426]
+ [-6.211496280528294e-47, 3.6607296186797117e-47, 0.7, 5.81547399618812e-47, -1.5891558132848647e-48, 0.7, -2.3283186142162194e-47, 1.8525166989190623e-47, 0.7, 1.7719350053561196e-47  …  0.32339181286549706, 1.3288096197339765e-45, -2.406573671857341e-45, 0.31754385964912274, -5.181053447216573e-46, 1.0122885108901054e-45, 0.31169590643274847, 1.5926495156599105e-46, -3.8101330106319815e-46, 0.30584795321637426]
+ [-2.325463438823169e-45, 1.4921950096721943e-45, 0.7, 2.1372135258244082e-45, -8.159199150016775e-47, 0.7, -8.674201303018941e-46, 7.407033154029957e-46, 0.7, 6.59121517028004e-46  …  0.32339181286549706, 4.681077546331119e-44, -9.168673246160421e-44, 0.31754385964912274, -1.8589372716338307e-44, 3.9259940822661885e-44, 0.31169590643274847, 5.853556290472661e-45, -1.5021289571420408e-44, 0.30584795321637426]
+ [-6.392817515195469e-44, 4.4001358292248024e-44, 0.7, 5.774217024408481e-44, -2.890251970346458e-45, 0.7, -2.374458385327122e-44, 2.1453565092353564e-44, 0.7, 1.80174698666066e-44  …  0.32339181286549706, 1.2132383431986734e-42, -2.532673744979001e-42, 0.31754385964912274, -4.904477697795913e-43, 1.1034157417570076e-42, 0.31169590643274847, 1.5807453384605294e-43, -4.289517343919539e-43, 0.30584795321637426]
+ [-1.3787503503174236e-42, 1.0073886474279489e-42, 0.7, 1.2251193474315923e-42, -7.708120549579181e-44, 0.7, -5.102248833206378e-43, 4.8300534112044794e-43, 0.7, 3.8666581968804824e-43  …  0.32339181286549706, 2.4704751359942366e-41, -5.44118362661684e-41, 0.31754385964912274, -1.0162156286973175e-41, 2.41103787004135e-41, 0.31169590643274847, 3.350572348324435e-42, -9.51981151113154e-42, 0.30584795321637426]
+ [-2.4329942701782505e-41, 1.8726572845802347e-41, 0.7, 2.1286192555782036e-41, -1.633848922522654e-42, 0.7, -8.97545663579173e-42, 8.838532404784254e-42, 0.7, 6.793904310090351e-42  …  0.32339181286549706, 4.120411791612597e-40, -9.504003130525254e-40, 0.31754385964912274, -1.7241821398234909e-40, 4.281941405290549e-40, 0.31169590643274847, 5.812886390913171e-41, -1.716752434217434e-40, 0.30584795321637426]
+ ⋮
+ [-0.002176856805330235, -0.005471508691304497, 0.7, -0.001099153144513662, -0.0029871877626791183, 0.7, -0.0005923337764093331, -0.0016674303283175998, 0.7, 0.0007288095726284696  …  0.32339181286549706, -0.0024465018058354463, -0.01637377384830735, 0.31754385964912274, -0.0016159836137553062, -0.01355179852781439, 0.31169590643274847, -0.0024066935622708515, -0.015385993847199006, 0.30584795321637426]
+ [-0.00250587547149793, -0.008500215882787284, 0.7, -0.0018252401083375962, -0.008052197913994155, 0.7, -0.0012170710172800508, -0.006710223227505286, 0.7, -0.000434683690218956  …  0.32339181286549706, -0.0021856746220264536, -0.01596720905195473, 0.31754385964912274, -0.001926403010673724, -0.013402096254005912, 0.31169590643274847, -0.002238819035873335, -0.014632665884650491, 0.30584795321637426]
+ [-0.0018622554232660391, -0.008813488862025372, 0.7, -0.0021123681485556273, -0.009894126107164069, 0.7, -0.002012840618838343, -0.009049578436370673, 0.7, -0.0018802163331253382  …  0.32339181286549706, -0.001421501025819743, -0.00996390185629101, 0.31754385964912274, -0.0019272324089571102, -0.012910684832531816, 0.31169590643274847, -0.0015105892195425286, -0.013900945522402237, 0.30584795321637426]
+ [-0.0011336758604551511, -0.006535539122471322, 0.7, -0.001979219507900437, -0.00762411437774105, 0.7, -0.0023947609189682666, -0.00818600309699748, 0.7, -0.0027194686723747835  …  0.32339181286549706, -0.00034224853703385805, -0.00457376401027034, 0.31754385964912274, -0.0011555137032241494, -0.011452364518071361, 0.31169590643274847, -0.0008953442781766002, -0.012890538549893608, 0.30584795321637426]
+ [-0.0008511358330743246, -0.003576478055447792, 0.7, -0.0015164577577977635, -0.0038311634821195746, 0.7, -0.0019345932262209958, -0.005644616064316158, 0.7, -0.002403265901442433  …  0.32339181286549706, 0.0004867695319956734, -0.0032409931754231974, 0.31754385964912274, -0.00013149800222093987, -0.00782656305647952, 0.31169590643274847, -0.0006596159967523298, -0.00951926804165878, 0.30584795321637426]
+ [-0.0007926983849506771, -0.0017254112742892866, 0.7, -0.0009183634647562862, -0.0015781479578827905, 0.7, -0.0009349159821019416, -0.00340549925301533, 0.7, -0.001330648411503606  …  0.32339181286549706, 0.0007085992217667344, -0.003227914524735306, 0.31754385964912274, 0.0003672721353326862, -0.002719344223435364, 0.31169590643274847, -0.0005220550795835429, -0.004148042975382867, 0.30584795321637426]
+ [-0.0005045949122517807, -0.0010626077342998173, 0.7, -0.00039778847670192055, -0.0013768855203502518, 0.7, -0.00012800109379730638, -0.00220010756935956, 0.7, -0.0004075931121329894  …  0.32339181286549706, 0.0005311117049730599, -0.0007705176971117157, 0.31754385964912274, 0.0002934328821294871, 0.0014618990740305471, 0.31169590643274847, -0.00014917439661154644, 0.0005201307641519544, 0.30584795321637426]
+ [7.803777012583456e-5, -0.0004293626682829821, 0.7, -4.958365526498939e-5, -0.0013994492190674875, 0.7, 8.636575719673748e-5, -0.0013855062117348852, 0.7, -0.00013921328594872933  …  0.32339181286549706, 0.0004477809010451735, 0.003410122319406037, 0.31754385964912274, 0.00020695567860444395, 0.0031178119822825095, 0.31169590643274847, 0.00038934073344127154, 0.002756152881021689, 0.30584795321637426]
+ [0.0006073616394551576, 0.000890705471058846, 0.7, 0.00018073121640876355, -8.677905121408308e-5, 0.7, -2.6115529692313e-5, -0.00015434992446148764, 0.7, -0.00025246394088185216  …  0.32339181286549706, 0.0006577431971269968, 0.005711584342450851, 0.31754385964912274, 0.00043018536651270086, 0.003066967628013788, 0.31169590643274847, 0.0007358879405425109, 0.003315270056768716, 0.30584795321637426]

After solving the equations, sol contains the solution for each of the three variables at every spatial point for each of the 100 points in time. The errors and integrals recorded by the AnalysisCallback can be obtained as NamedTuples by errors(analysis_callback) and integrals(analysis_callback).

Visualize results

After running the simulation, the results can be visualized using Plots.jl, which needs to be imported first. Then, we can plot the solution at the final time by calling plot on a Pair of the Semidiscretization and the corresponding ODESolution sol. The result is depicted in the following picture.

using Plots
+
+plot(semi => sol)

shoaling solution

By default, this will plot the bathymetry, but not the initial (analytical) solution. You can adjust this by passing the boolean values plot_bathymetry (if true always plot to first subplot) and plot_initial. You can also provide a conversion function that converts the solution. A conversion function should take the values of the primitive variables q at one node, and the equations as input and should return an SVector of any length as output. For a user defined conversion function, there should also exist a function varnames(conversion, equations) that returns a Tuple of the variable names used for labelling. The conversion function can, e.g., be prim2cons or waterheight_total if one only wants to plot the total waterheight. The resulting plot will have one subplot for each of the returned variables of the conversion variable. By default, the conversion function is just prim2prim, i.e. the identity.

Plotting an animation over time can, e.g., be done by the following command, which uses step to plot the solution at a specific time step.

anim = @animate for step in 1:length(sol.u)
+    plot(semi => sol, plot_initial = true, conversion = waterheight_total, step = step, xlim = (-50, 20), ylims = (-0.8, 0.1))
+end
+gif(anim, "shoaling_solution.gif", fps = 25)
[ Info: Saved animation to /home/runner/work/DispersiveShallowWater.jl/DispersiveShallowWater.jl/docs/build/shoaling_solution.gif

shoaling solution

It is also possible to plot the solution variables at a fixed spatial point over time by calling plot(semi => sol, x) for some x-value, see plot_examples.jl from the reproducibility repository of the master thesis of Joshua Lampert for some examples.

Often, it is interesting to have a look at how the quantities that are recorded by the AnalysisCallback evolve in time. To this end, you can plot the AnalysisCallback by

plot(analysis_callback)

This creates the following figure:

analysis callback

You can see that the linear invariants $\int_\Omega\eta\textrm{d}x$ and $\int_\Omega v\textrm{d}x$ are indeed conserved exactly. The entropy, however, starts growing at around $t = 17$ and rises up to approximately $5e-5$. This is because of the fact that, during the time integration, a nonlinear invariant is not necessarily conserved, even if the semidiscretization conserves the quantity exactly. How to obtain a fully-discrete structure-preserving numerical scheme is explained in the following section.

Use entropy-conserving time integration

To obtain entropy-conserving time-stepping schemes DispersiveShallowWater.jl uses the relaxation method introduced in [Ketcheson2019] and further developed in [RanochaSayyariDalcinParsaniKetcheson2020]. The relaxation method is implemented as a RelaxationCallback, which takes a function representing the conserved quantity as the keyword argument invariant. Therefore, we can run the same example as above, but using relaxation on the entropy by simply adding another callback to the CallbackSet:

analysis_callback = AnalysisCallback(semi; interval = 10,
+                                     extra_analysis_errors = (:conservation_error,),
+                                     extra_analysis_integrals = (waterheight_total,
+                                                                 velocity, entropy),
+                                     io = devnull)
+relaxation_callback = RelaxationCallback(invariant = entropy)
+callbacks = CallbackSet(relaxation_callback, analysis_callback)
+sol = solve(ode, Tsit5(), abstol = 1e-7, reltol = 1e-7,
+            save_everystep = false, callback = callbacks, saveat = saveat)
retcode: Success
+Interpolation: 1st order linear
+t: 100-element Vector{Float64}:
+  0.0
+  0.25252525252525254
+  0.5050505050505051
+  0.7575757575757576
+  1.0101010101010102
+  1.2626262626262625
+  1.5151515151515151
+  1.7676767676767677
+  2.0202020202020203
+  2.272727272727273
+  ⋮
+ 22.97979797979798
+ 23.232323232323232
+ 23.484848484848484
+ 23.737373737373737
+ 23.98989898989899
+ 24.242424242424242
+ 24.494949494949495
+ 24.747474747474747
+ 25.0
+u: 100-element Vector{Vector{Float64}}:
+ [0.0, 0.0, 0.7, 0.0, 0.0, 0.7, 0.0, 0.0, 0.7, 0.0  …  0.32339181286549706, 1.1159911034969113e-106, 0.0, 0.31754385964912274, 6.2568988186290324e-108, 0.0, 0.31169590643274847, 3.4485093610035348e-109, 0.0, 0.30584795321637426]
+ [-3.332266410556691e-56, 8.250693938128724e-57, 0.7, 3.407583642309716e-56, 3.8490501278015398e-59, 0.7, -1.2832354556318063e-56, 4.5801232603284077e-57, 0.7, 9.844890923810958e-57  …  0.32339181286549706, 9.30670509753227e-55, -7.283825400163255e-55, 0.31754385964912274, -3.350861542354207e-55, 2.838655119185712e-55, 0.31169590643274847, 9.268574447724803e-56, -9.95620229491928e-56, 0.30584795321637426]
+ [-4.854545545070253e-53, 1.8097193833399515e-53, 0.7, 4.842352321849954e-53, -1.4595167657656438e-55, 0.7, -1.8543498712251333e-53, 9.787486255308402e-54, 0.7, 1.4192559489927457e-53  …  0.32339181286549706, 1.2618974617562503e-51, -1.4764810496891982e-51, 0.31754385964912274, -4.641297573080381e-52, 5.871219778928541e-52, 0.31169590643274847, 1.3208456304778785e-52, -2.098129111059335e-52, 0.30584795321637426]
+ [-1.161174020717534e-50, 5.352294982928633e-51, 0.7, 1.1319863540772265e-50, -1.0908116767888405e-52, 0.7, -4.4038264457122423e-51, 2.8256156036521797e-51, 0.7, 3.3633753287785445e-51  …  0.32339181286549706, 2.818771303094653e-49, -4.048100076844979e-49, 0.31754385964912274, -1.0581360622684687e-49, 1.6415282643014427e-49, 0.31169590643274847, 3.0939972995600496e-50, -5.973308375246117e-50, 0.30584795321637426]
+ [-1.1119340370914712e-48, 5.886606126151948e-49, 0.7, 1.0619051193616636e-48, -1.8734919750652443e-50, 0.7, -4.1913262441970204e-49, 3.0415055830255834e-49, 0.7, 3.195203669716197e-49  …  0.32339181286549706, 2.533534775326589e-47, -4.151029387758137e-47, 0.31754385964912274, -9.692959041345168e-48, 1.7142717056808075e-47, 0.31169590643274847, 2.906199272580485e-48, -6.343937197528971e-48, 0.30584795321637426]
+ [-6.211342952679108e-47, 3.660636442561326e-47, 0.7, 5.815331336869501e-47, -1.5891123999359684e-48, 0.7, -2.3282612408209723e-47, 1.852469822382361e-47, 0.7, 1.7718913654148852e-47  …  0.32339181286549706, 1.3287774802196427e-45, -2.4065136543212134e-45, 0.31754385964912274, -5.180927344176877e-46, 1.0122631249018097e-45, 0.31169590643274847, 1.592610437929165e-46, -3.8100369833174064e-46, 0.30584795321637426]
+ [-2.3254164502674176e-45, 1.49216372070005e-45, 0.7, 2.137170722072698e-45, -8.159011679159638e-47, 0.7, -8.674026426679073e-46, 7.406879206110161e-46, 0.7, 6.591082384100245e-46  …  0.32339181286549706, 4.68098574598978e-44, -9.168487035994518e-44, 0.31754385964912274, -1.8589004819059543e-44, 3.9259136618095353e-44, 0.31169590643274847, 5.853439066560083e-45, -1.502097946470543e-44, 0.30584795321637426]
+ [-6.392705132782543e-44, 4.400056006120195e-44, 0.7, 5.774116364831327e-44, -2.890195404164474e-45, 0.7, -2.3744167260793885e-44, 2.1453179125315753e-44, 0.7, 1.8017153964783695e-44  …  0.32339181286549706, 1.213217630459334e-42, -2.5326292145530362e-42, 0.31754385964912274, -4.904393213034871e-43, 1.1033961767416291e-42, 0.31169590643274847, 1.5807177915108583e-43, -4.289440699461812e-43, 0.30584795321637426]
+ [-1.3787258559015014e-42, 1.0073701524772098e-42, 0.7, 1.2250977911673285e-42, -7.707967242881186e-44, 0.7, -5.102158372622728e-43, 4.8299655820378e-43, 0.7, 3.866589693374949e-43  …  0.32339181286549706, 2.470432749855349e-41, -5.441087426494035e-41, 0.31754385964912274, -1.0161980065048432e-41, 2.410994807691375e-41, 0.31169590643274847, 3.3505134403765806e-42, -9.51963989884522e-42, 0.30584795321637426]
+ [-2.432953367861249e-41, 1.8726248798384983e-41, 0.7, 2.1285837948336777e-41, -1.6338186152122516e-42, 0.7, -8.975306003526185e-42, 8.83838082751108e-42, 0.7, 6.79379036762082e-42  …  0.32339181286549706, 4.12034484685199e-40, -9.503844614433176e-40, 0.31754385964912274, -1.7241538314594997e-40, 4.2818692725079266e-40, 0.31169590643274847, 5.812789657041895e-41, -1.7167232499767342e-40, 0.30584795321637426]
+ ⋮
+ [-0.0021659838088561004, -0.005462316236984093, 0.7, -0.001094995519112734, -0.0029946780624175644, 0.7, -0.0005920726738234917, -0.0016630027332406104, 0.7, 0.0007264876839320953  …  0.32339181286549706, -0.002445583550959626, -0.016317015455508944, 0.31754385964912274, -0.0016212354480480975, -0.013528910892895053, 0.31169590643274847, -0.002396885241065678, -0.015332363345197464, 0.30584795321637426]
+ [-0.0024946509575851233, -0.0084743050248458, 0.7, -0.0018211108261936442, -0.008025725779320278, 0.7, -0.0012193167641073581, -0.006688548166333666, 0.7, -0.00043839507595223084  …  0.32339181286549706, -0.002180430239348519, -0.01590186318296433, 0.31754385964912274, -0.0019262159652891774, -0.013401193628169689, 0.31169590643274847, -0.002232448416912339, -0.014629354103318216, 0.30584795321637426]
+ [-0.0018597237172830337, -0.008785908167303248, 0.7, -0.002108029863372081, -0.009851920630150779, 0.7, -0.002009838115979129, -0.009018710861238729, 0.7, -0.0018764289682747922  …  0.32339181286549706, -0.0014174867196701073, -0.00995515983630038, 0.31754385964912274, -0.0019200772426444456, -0.012904053282721581, 0.31169590643274847, -0.0015113103521349474, -0.013907495809161828, 0.30584795321637426]
+ [-0.0011382887415978525, -0.006526240914208616, 0.7, -0.00197556375734427, -0.007603373094278387, 0.7, -0.002386647597983837, -0.00816566395304658, 0.7, -0.002708923707548109  …  0.32339181286549706, -0.00034409204632861916, -0.004613565382136222, 0.31754385964912274, -0.001150846246748621, -0.011431378783413654, 0.31169590643274847, -0.0008995601062787052, -0.012874967436299362, 0.30584795321637426]
+ [-0.0008547943895035529, -0.0035873298420345387, 0.7, -0.0015148921303084792, -0.003843847009732438, 0.7, -0.0019284466945888887, -0.005644129387757395, 0.7, -0.0023946513342133087  …  0.32339181286549706, 0.00048076743678616995, -0.003265547193722709, 0.31754385964912274, -0.00013500790692120215, -0.007808778243089925, 0.31169590643274847, -0.0006613057774831114, -0.009496236940962929, 0.30584795321637426]
+ [-0.0007911392786587085, -0.001739356955529783, 0.7, -0.0009192247258541631, -0.0016012733935531107, 0.7, -0.0009360248152808343, -0.0034159764533118787, 0.7, -0.0013305081833949845  …  0.32339181286549706, 0.0007042413951192331, -0.0032091589879662194, 0.31754385964912274, 0.00036098704059875846, -0.002724084533864268, 0.31169590643274847, -0.0005197291955477037, -0.004146557210079549, 0.30584795321637426]
+ [-0.0005009119017044855, -0.0010643539943988355, 0.7, -0.0003997104601284641, -0.0013829707632916784, 0.7, -0.00013392608631431288, -0.0022067354929193777, 0.7, -0.00041367937266736314  …  0.32339181286549706, 0.0005317240886544872, -0.0007458554006874356, 0.31754385964912274, 0.0002918660500146939, 0.0014385389505501853, 0.31169590643274847, -0.00014667287237832502, 0.000499635832292766, 0.30584795321637426]
+ [7.877033748192985e-5, -0.000421123639625422, 0.7, -5.0759247674148775e-5, -0.0013861889826008653, 0.7, 8.248268091041952e-5, -0.0013829603946880502, 0.7, -0.00014394433252072245  …  0.32339181286549706, 0.0004509502003486085, 0.0034010197767588668, 0.31754385964912274, 0.0002098810724059527, 0.0030986111294803113, 0.31169590643274847, 0.0003882810307982635, 0.0027368433078829644, 0.30584795321637426]
+ [0.0006041217258656232, 0.0008950650498073326, 0.7, 0.0001807892600773303, -7.445962324119322e-5, 0.7, -2.4516528465226556e-5, -0.0001493914009533822, 0.7, -0.0002513925977371086  …  0.32339181286549706, 0.0006585897415157567, 0.0056799234308563176, 0.31754385964912274, 0.0004319778826393965, 0.003066796203846808, 0.31169590643274847, 0.0007320680612230762, 0.0033111405751278435, 0.30584795321637426]

When you use both, an AnalysisCallback and a RelaxationCallback, note that the relaxation_callback needs to come first inside the CallbackSet as it needs to be invoked prior to the analysis_callback, such that the analysis_callback analyzes the solution with the already updated values.

Plotting the analysis_callback again, we can see that now also the entropy is conserved up to machine precision.

plot(analysis_callback, ylims = (-5e-16, 5e-16))

analysis callback relaxation

Customize solver

In the semidiscretization created above, we used the default SBP operators, which are periodic finite difference operators. Using different SBP operators for the semidiscretization can be done leveraging SummationByPartsOperators.jl, which needs to be imported first:

using SummationByPartsOperators: legendre_derivative_operator, UniformPeriodicMesh1D, couple_discontinuously, PeriodicUpwindOperators

As an example, let us create a semidiscretization based on discontinuous Galerkin (DG) upwind operators. A semidiscretization implemented in DispersiveShallowWater.jl needs one first-derivative and one second-derivative SBP operator. To build the first-derivative operator, we first create a LegendreDerivativeOperator with polynomial degree 3 on a reference element [-1.0, 1.0] and a UniformPeriodicMesh1D for the coupling.

mesh = Mesh1D(coordinates_min, coordinates_max, N)
+accuracy_order = 4
+D_legendre = legendre_derivative_operator(-1.0, 1.0, accuracy_order)
+uniform_mesh = UniformPeriodicMesh1D(mesh.xmin, mesh.xmax, div(mesh.N, accuracy_order))
SummationByPartsOperators.UniformPeriodicMesh1D{Float64} with 128 cells in (-130.0, 20.0)

Upwind DG operators in negative, central and positive operators can be obtained by couple_discontinuously

central = couple_discontinuously(D_legendre, uniform_mesh)
+minus = couple_discontinuously(D_legendre, uniform_mesh, Val(:minus))
+plus = couple_discontinuously(D_legendre, uniform_mesh, Val(:plus))
+D1 = PeriodicUpwindOperators(minus, central, plus)
Upwind SBP first-derivative operators of order 3 on a grid in [-129.99999999999997, 19.999999999999996] using 512 nodes 
+and coefficients of Module

In order to still have an entropy-conserving semidiscretization the second-derivative SBP operator needs to be

using SparseArrays: sparse
+D2 = sparse(plus) * sparse(minus)
512×512 SparseArrays.SparseMatrixCSC{Float64, Int64} with 3072 stored entries:
+⎡⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⎤
+⎢⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⎥
+⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⎥
+⎣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⎦

The Solver object can now be created by passing the two SBP operators to the constructor, which, in turn, can be used to construct a Semidiscretization:

solver = Solver(D1, D2)
+semi = Semidiscretization(mesh, equations, initial_condition, solver, boundary_conditions = boundary_conditions)
Semidiscretization
+    #spatial dimensions: 1
+    mesh: Mesh1D{Float64} with length 512
+    equations: BBMBBMVariableEquations1D
+    initial condition: initial_condition_shoaling
+    boundary condition: boundary_condition_periodic
+    source terms: nothing

As before, we can run the simulation by

analysis_callback = AnalysisCallback(semi; interval = 10,
+                                     extra_analysis_errors = (:conservation_error,),
+                                     extra_analysis_integrals = (waterheight_total,
+                                                                 velocity, entropy),
+                                     io = devnull)
+relaxation_callback = RelaxationCallback(invariant = entropy)
+callbacks = CallbackSet(relaxation_callback, analysis_callback)
+sol = solve(ode, Tsit5(), abstol = 1e-7, reltol = 1e-7,
+            save_everystep = false, callback = callbacks, saveat = saveat)
+anim = @animate for step in 1:length(sol.u)
+    plot(semi => sol, plot_initial = true, conversion = waterheight_total, step = step, xlim = (-50, 20), ylims = (-0.8, 0.1))
+end
+gif(anim, "shoaling_solution_dg.gif", fps = 25)
[ Info: Saved animation to /home/runner/work/DispersiveShallowWater.jl/DispersiveShallowWater.jl/docs/build/shoaling_solution_dg.gif

shoaling solution DG

For more details see also the documentation of SummationByPartsOperators.jl

Additional resources

Some more examples sorted by the simulated equations can be found in the examples/ subdirectory. Especially, in examples/svaerd_kalisch_1d/ you can find Julia scripts that solve the SvaerdKalischEquations1D that were not covered in this tutorial. The same steps as described above, however, apply in the same way to these equations. Attention must be paid for these equations because they do not conserve the classical total entropy $\mathcal E$, but a modified entropy $\hat{\mathcal E}$, available as entropy_modified.

More examples, especially focussing on plotting, can be found in the scripts create_figures.jl and plot_examples.jl from the reproducibility repository of the master thesis of Joshua Lampert.

References

diff --git a/v0.3.2/ref-trixibase/index.html b/v0.3.2/ref-trixibase/index.html new file mode 100644 index 00000000..d176ad79 --- /dev/null +++ b/v0.3.2/ref-trixibase/index.html @@ -0,0 +1,10 @@ + +TrixiBase · DispersiveShallowWater.jl
GitHub

TrixiBase.jl API

TrixiBase.trixi_include — Method
trixi_include([mod::Module=Main,] elixir::AbstractString; kwargs...)

include the file elixir and evaluate its content in the global scope of module mod. You can override specific assignments in elixir by supplying keyword arguments. Its basic purpose is to make it easier to modify some parameters while running simulations from the REPL. Additionally, this is used in tests to reduce the computational burden for CI while still providing examples with sensible default values for users.

Before replacing assignments in elixir, the keyword argument maxiters is inserted into calls to solve with it's default value used in the SciML ecosystem for ODEs, see the "Miscellaneous" section of the documentation.

Examples

julia> using TrixiBase, Trixi
+
+julia> redirect_stdout(devnull) do
+         trixi_include(@__MODULE__, joinpath(examples_dir(), "tree_1d_dgsem", "elixir_advection_extended.jl"),
+                       tspan=(0.0, 0.1))
+         sol.t[end]
+       end
+[ Info: You just called `trixi_include`. Julia may now compile the code, please be patient.
+0.1
source
diff --git a/v0.3.2/ref/index.html b/v0.3.2/ref/index.html new file mode 100644 index 00000000..145b865c --- /dev/null +++ b/v0.3.2/ref/index.html @@ -0,0 +1,24 @@ + +DispersiveShallowWater · DispersiveShallowWater.jl
GitHub

DispersiveShallowWater.jl API

DispersiveShallowWater.AbstractEquations — Type
AbstractEquations{NDIMS, NVARS}

An abstract supertype of specific equations such as the BBM-BBM equations. The type parameters encode the number of spatial dimensions (NDIMS) and the number of primary variables (NVARS) of the physics model.

source
DispersiveShallowWater.AnalysisCallback — Type
AnalysisCallback(semi; interval=0,
+                       extra_analysis_errors=Symbol[],
+                       extra_analysis_integrals=(),
+                       io=stdout)

Analyze a numerical solution every interval time steps. The L2- and the L∞-norm for each component are computed by default. Additional errors can be computed, e.g. by passing extra_analysis_errors = (:conservation_error,).

Further scalar functions func in extra_analysis_integrals are applied to the numerical solution and integrated over the computational domain. Some examples for this are entropy, and energy_total. You can also write your own function with the same signature as the examples listed above and pass it via extra_analysis_integrals. The computed errors and intergrals are saved for each timestep and can be obtained by calling errors and integrals.

During the Simulation, the AnalysisCallback will print information to io.

source
DispersiveShallowWater.BBMBBMEquations1D — Type
BBMBBMEquations1D(gravity, D, eta0 = 0.0)

BBM-BBM (Benjamin–Bona–Mahony) system in one spatial dimension. The equations are given by

\[\begin{aligned} + \eta_t + ((\eta + D)v)_x - \frac{1}{6}D^2\eta_{xxt} &= 0,\\ + v_t + g\eta_x + \left(\frac{1}{2}v^2\right)_x - \frac{1}{6}D^2v_{xxt} &= 0. +\end{aligned}\]

The unknown quantities of the BBM-BBM equations are the total water height $\eta$ and the velocity $v$. The gravitational constant is denoted by g and the constant bottom topography (bathymetry) $b = \eta_0 - D$. The water height above the bathymetry is therefore given by $h = \eta - \eta_0 + D$. The BBM-BBM equations are only implemented for $\eta_0 = 0$.

One reference for the BBM-BBM system can be found in

source
DispersiveShallowWater.BBMBBMVariableEquations1D — Type
BBMBBMVariableEquations1D(gravity, eta0 = 1.0)

BBM-BBM (Benjamin–Bona–Mahony) system in one spatial dimension with spatially varying bathymetry. The equations are given by

\[\begin{aligned} + \eta_t + ((\eta + D)v)_x - \frac{1}{6}(D^2\eta_{xt})_x &= 0,\\ + v_t + g\eta_x + \left(\frac{1}{2}v^2\right)_x - \frac{1}{6}(D^2v_t)_{xx} &= 0. +\end{aligned}\]

The unknown quantities of the BBM-BBM equations are the total water height $\eta$ and the velocity $v$. The gravitational constant is denoted by g and the bottom topography (bathymetry) $b = \eta_0 - D$. The water height above the bathymetry is therefore given by $h = \eta - \eta_0 + D$. The BBM-BBM equations are only implemented for $\eta_0 = 0$.

One reference for the BBM-BBM system with spatially varying bathymetry can be found in

  • Samer Israwi, Henrik Kalisch, Theodoros Katsaounis, Dimitrios Mitsotakis (2022) A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis DOI: 10.1088/1361-6544/ac3c29
source
DispersiveShallowWater.RelaxationCallback — Type
RelaxationCallback(invariant)

Use a relaxation method in time in order to exactly preserve the (nonlinear) invariant for a conservative semidiscretization. A possible choice for invariant is invariant = entropy.

Reference

  • Hendrik Ranocha, Mohammed Sayyari, Lisandro Dalcin, Matteo Parsani, David I. Ketcheson (2020) Relaxation Runge–Kutta Methods: Fully-Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier–Stokes Equations DOI: 10.1137/19M1263480
source
DispersiveShallowWater.Semidiscretization — Method
Semidiscretization(mesh, equations, initial_condition, solver;
+                   source_terms=nothing,
+                   boundary_conditions=boundary_condition_periodic,
+                   RealT=real(solver),
+                   uEltype=RealT,
+                   initial_cache=NamedTuple())

Construct a semidiscretization of a PDE.

source
DispersiveShallowWater.Solver — Method
Solver(mesh, accuracy_order)

Create a solver, where the summation by parts (SBP) operators are of order accuracy_order and associated to the mesh.

source
DispersiveShallowWater.Solver — Method
Solver(D1, D2)

Create a solver, where D1 is an AbstractDerivativeOperator from SummationByPartsOperators.jl of first derivative_order and D2 is an AbstractDerivativeOperator of second derivative_order or an AbstractMatrix. Both summation by parts operators should be associated with the same grid.

source
DispersiveShallowWater.SummaryCallback — Type
SummaryCallback(io::IO = stdout)

Create and return a callback that resets the timer at the beginning of a simulation and prints the timer values at the end of the simulation.

source
DispersiveShallowWater.SvaerdKalischEquations1D — Type
SvärdKalischEquations1D(gravity, eta0 = 1.0, alpha = 0.0, beta = 0.2308939393939394, gamma = 0.04034343434343434)

Dispersive system by Svärd and Kalisch in one spatial dimension with spatially varying bathymetry. The equations are given in conservative variables by

\[\begin{aligned} + h_t + (hv)_x &= (\hat\alpha(\hat\alpha(h + b)_x)_x)_x,\\ + (hv)_t + (hv^2)_x + gh(h + b)_x &= (\hat\alpha v(\hat\alpha(h + b)_x)_x)_x + (\hat\beta v_x)_{xt} + \frac{1}{2}(\hat\gamma v_x)_{xx} + \frac{1}{2}(\hat\gamma v_{xx})_x, +\end{aligned}\]

where $\hat\alpha^2 = \alpha\sqrt{gD}D^2$, $\hat\beta = \beta D^3$, $\hat\gamma = \gamma\sqrt{gD}D^3$. The coefficients $\alpha$, $\beta$ and $\gamma$ are provided in dimensionless form and $D = \eta_0 - b$ is the still-water depth and eta0 is the still-water surface (lake-at-rest). The equations can be rewritten in primitive variables as

\[\begin{aligned} + \eta_t + ((\eta + D)v)_x = (\hat\alpha(\hat\alpha\eta_x)_x)_x,\\ + v_t(\eta + D) - v((\eta + D)v)_x + ((\eta + D)v^2)_x + g(\eta + D)\eta_x &= (\hat\alpha v(\hat\alpha\eta_x)_x)_x - v(\hat\alpha(\hat\alpha\eta_x)_x)_x + (\hat\beta v_x)_{xt} + \frac{1}{2}(\hat\gamma v_x)_{xx} + \frac{1}{2}(\hat\gamma v_{xx})_x. +\end{aligned}\]

The unknown quantities of the Svärd-Kalisch equations are the total water height $\eta$ and the velocity $v$. The gravitational constant is denoted by g and the bottom topography (bathymetry) $b = \eta_0 - D$. The water height above the bathymetry is therefore given by $h = \eta - \eta_0 + D$.

The equations by Svärd and Kalisch are presented and analyzed in

  • Magnus Svärd, Henrik Kalisch (2023) A novel energy-bounded Boussinesq model and a well-balanced and stable numerical discretization arXiv: 2302.09924
source
DispersiveShallowWater.cons2prim — Function
cons2prim(u, equations)

Convert the conserved variables u to the primitive variables for a given set of equations. u is a vector type of the correct length nvariables(equations). Notice the function doesn't include any error checks for the purpose of efficiency, so please make sure your input is correct. The inverse conversion is performed by prim2cons.

source
DispersiveShallowWater.convergence_test — Method
convergence_test([mod::Module=Main,] example::AbstractString, iterations; kwargs...)

Run iterations simulations using the setup given in example and compute the experimental order of convergence (EOC) in the $L^2$ and $L^\infty$ norm. In each iteration, the resolution of the respective mesh will be doubled. Additional keyword arguments kwargs... and the optional module mod are passed directly to trixi_include.

Adjusted from Trixi.jl.

source
DispersiveShallowWater.eachvariable — Method
eachvariable(equations::AbstractEquations)

Return an iterator over the indices that specify the location in relevant data structures for the variables in equations. In particular, not the variables themselves are returned.

source
DispersiveShallowWater.energy_total — Function
energy_total(q, equations)

Return the total energy of the primitive variables q for a given set of equations.

q is a vector of the primitive variables at a single node, i.e., a vector of the correct length nvariables(equations).

source
DispersiveShallowWater.energy_total_modified — Method
energy_total_modified(q, equations::SvaerdKalischEquations1D, cache)

Return the modified total energy of the primitive variables q for the SvaerdKalischEquations1D. It contains an additional term containing a derivative compared to the usual energy_total. The energy_total_modified is a conserved quantity of the Svärd-Kalisch equations.

q is a vector of the primitive variables at ALL nodes, i.e., a matrix of the correct length nvariables(equations) as first dimension and the number of nodes as length of the second dimension. cache needs to hold the first-derivative SBP operator D1.

source
DispersiveShallowWater.entropy — Function
entropy(q, equations)

Return the entropy of the primitive variables q for a given set of equations.

q is a vector of the primitive variables at a single node, i.e., a vector of the correct length nvariables(equations).

source
DispersiveShallowWater.errors — Method
errors(analysis_callback)

Return the computed errors for each timestep as a named tuple. The shape of each entry is (nvariables, ntimesteps).

source
DispersiveShallowWater.examples_dir — Method
examples_dir()

Return the directory where the example files provided with DispersiveShallowWater.jl are located. If DispersiveShallowWater is installed as a regular package (with ]add DispersiveShallowWater), these files are read-only and should not be modified. To find out which files are available, use, e.g., readdir.

Copied from Trixi.jl.

Examples

readdir(examples_dir())
source
DispersiveShallowWater.get_name — Method
get_name(equations::AbstractEquations)

Return the canonical, human-readable name for the given system of equations.

Examples

julia> DispersiveShallowWater.get_name(BBMBBMEquations1D(gravity_constant=1.0))
+"BBMBBMEquations1D"
source
DispersiveShallowWater.initial_condition_dingemans — Method
initial_condition_dingemans(x, t, equations::BBMBBMVariableEquations1D, mesh)

The initial condition that uses the dispersion relation of the Euler equations to approximate waves generated by a wave maker as it is done by experiments of Dingemans. The topography is a trapezoidal.

Translation of water height

The initial condition for the water height is translated to be around 0, which is needed for the simulation because the BBMBBMVariableEquations1D are only implemented for $ta_0 = 0$.

References:

  • Magnus Svärd, Henrik Kalisch (2023) A novel energy-bounded Boussinesq model and a well-balanced and stable numerical discretization arXiv: 2302.09924
  • Maarten W. Dingemans (1994) Comparison of computations with Boussinesq-like models and laboratory measurements link
source
DispersiveShallowWater.initial_condition_dingemans — Method
initial_condition_dingemans(x, t, equations::SvaerdKalischEquations1D, mesh)

The initial condition that uses the dispersion relation of the Euler equations to approximate waves generated by a wave maker as it is done by experiments of Dingemans. The topography is a trapezoidal. It is assumed that equations.eta0 = 0.8.

References:

  • Magnus Svärd, Henrik Kalisch (2023) A novel energy-bounded Boussinesq model and a well-balanced and stable numerical discretization arXiv: 2302.09924
  • Maarten W. Dingemans (1994) Comparison of computations with Boussinesq-like models and laboratory measurements link
source
DispersiveShallowWater.momentum — Method
momentum(q, equations)

Return the momentum of the primitive variables q for a given set of equations, i.e. the waterheight times the velocity.

q is a vector of the primitive variables at a single node, i.e., a vector of the correct length nvariables(equations).

source
DispersiveShallowWater.prim2cons — Function
prim2cons(q, equations)

Convert the primitive variables q to the conserved variables for a given set of equations. q is a vector type of the correct length nvariables(equations). Notice the function doesn't include any error checks for the purpose of efficiency, so please make sure your input is correct. The inverse conversion is performed by cons2prim.

source
DispersiveShallowWater.varnames — Function
varnames(conversion_function, equations)

Return the list of variable names when applying conversion_function to the conserved variables associated to equations. Common choices of the conversion_function are prim2prim and prim2cons.

source
DispersiveShallowWater.velocity — Function
velocity(q, equations)

Return the velocity of the primitive variables q for a given set of equations.

q is a vector of the primitive variables at a single node, i.e., a vector of the correct length nvariables(equations).

source
DispersiveShallowWater.waterheight — Function
waterheight(q, equations)

Return the waterheight of the primitive variables q for a given set of equations, i.e. the waterheight above the bathymetry.

q is a vector of the primitive variables at a single node, i.e., a vector of the correct length nvariables(equations).

source
DispersiveShallowWater.waterheight_total — Function
waterheight_total(q, equations)

Return the total waterheight of the primitive variables q for a given set of equations, i.e. the waterheight plus the bathymetry.

q is a vector of the primitive variables at a single node, i.e., a vector of the correct length nvariables(equations).

source
DispersiveShallowWater.@autoinfiltrate — Macro
@autoinfiltrate
+@autoinfiltrate condition::Bool

Invoke the @infiltrate macro of the package Infiltrator.jl to create a breakpoint for ad-hoc interactive debugging in the REPL. If the optional argument condition is given, the breakpoint is only enabled if condition evaluates to true.

As opposed to using Infiltrator.@infiltrate directly, this macro does not require Infiltrator.jl to be added as a dependency to DispersiveShallowWater.jl. As a bonus, the macro will also attempt to load the Infiltrator module if it has not yet been loaded manually.

Note: For this macro to work, the Infiltrator.jl package needs to be installed in your current Julia environment stack.

See also: Infiltrator.jl

Internal use only

Please note that this macro is intended for internal use only. It is not part of the public API of DispersiveShallowWater.jl, and it thus can altered (or be removed) at any time without it being considered a breaking change.

source
diff --git a/v0.3.2/search_index.js b/v0.3.2/search_index.js new file mode 100644 index 00000000..dea2bdef --- /dev/null +++ b/v0.3.2/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"ref-trixibase/#TrixiBase.jl-API","page":"TrixiBase","title":"TrixiBase.jl API","text":"","category":"section"},{"location":"ref-trixibase/","page":"TrixiBase","title":"TrixiBase","text":"CurrentModule = TrixiBase","category":"page"},{"location":"ref-trixibase/","page":"TrixiBase","title":"TrixiBase","text":"Modules = [TrixiBase]","category":"page"},{"location":"ref-trixibase/#TrixiBase.trixi_include-Tuple{Module, AbstractString}","page":"TrixiBase","title":"TrixiBase.trixi_include","text":"trixi_include([mod::Module=Main,] elixir::AbstractString; kwargs...)\n\ninclude the file elixir and evaluate its content in the global scope of module mod. You can override specific assignments in elixir by supplying keyword arguments. Its basic purpose is to make it easier to modify some parameters while running simulations from the REPL. Additionally, this is used in tests to reduce the computational burden for CI while still providing examples with sensible default values for users.\n\nBefore replacing assignments in elixir, the keyword argument maxiters is inserted into calls to solve with it's default value used in the SciML ecosystem for ODEs, see the \"Miscellaneous\" section of the documentation.\n\nExamples\n\njulia> using TrixiBase, Trixi\n\njulia> redirect_stdout(devnull) do\n trixi_include(@__MODULE__, joinpath(examples_dir(), \"tree_1d_dgsem\", \"elixir_advection_extended.jl\"),\n tspan=(0.0, 0.1))\n sol.t[end]\n end\n[ Info: You just called `trixi_include`. Julia may now compile the code, please be patient.\n0.1\n\n\n\n\n\n","category":"method"},{"location":"license/#License","page":"License","title":"License","text":"","category":"section"},{"location":"license/","page":"License","title":"License","text":"MIT License","category":"page"},{"location":"license/","page":"License","title":"License","text":"Copyright (c) 2023-present Joshua Lampert and contributors","category":"page"},{"location":"license/","page":"License","title":"License","text":"Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the \"Software\"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:","category":"page"},{"location":"license/","page":"License","title":"License","text":"The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.","category":"page"},{"location":"license/","page":"License","title":"License","text":"THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.","category":"page"},{"location":"development/#Development","page":"Development","title":"Development","text":"","category":"section"},{"location":"development/","page":"Development","title":"Development","text":"If you have any suggestions or ideas for improvements or new features, we are pleased to accept and discuss issues or if you are willing to contribute, feel free to open a pull request, even if it is only fixing a typo or improving the docs.","category":"page"},{"location":"development/#Changing-DispersiveShallowWater.jl-and-running-it-locally","page":"Development","title":"Changing DispersiveShallowWater.jl and running it locally","text":"","category":"section"},{"location":"development/","page":"Development","title":"Development","text":"If you plan to edit DispersiveShallowWater.jl, you first need to clone a local copy of the repository, which can be done by using git. It is recommended that you create a project, e.g. call it run, inside the repository, where you can add packages that you use during executing and testing DispersiveShallowWater.jl, but are not needed by DispersiveShallowWater.jl. This way you can keep the Project.toml of the main repository clean. To do so, you can execute the following lines in a terminal:","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"git clone https://github.com/JoshuaLampert/DispersiveShallowWater.jl.git\ncd DispersiveShallowWater\nmkdir run\ncd run\njulia --project=. -e 'using Pkg; Pkg.develop(PackageSpec(path=\"..\"))' # Install local DispersiveShallowWater.jl clone\njulia --project=. -e 'using Pkg; Pkg.add([\"OrdinaryDiffEq\", \"Plots\", \"SummationByPartsOperators\"])' # Install additional packages","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"If you use other packages for executing DispersiveShallowWater.jl, you can add them to the project in the run directory in an analogous way as above. To use the Julia project within run, be sure to start the Julia REPL by","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"julia --project=.","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"if already inside the the run directory or julia --project=run if in the main directory of the repo.","category":"page"},{"location":"development/#Preview-of-the-documentation","page":"Development","title":"Preview of the documentation","text":"","category":"section"},{"location":"development/","page":"Development","title":"Development","text":"If you want to build the documentation locally, you can run","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"julia --project=docs -e 'using Pkg; Pkg.develop(PackageSpec(path=pwd())); Pkg.instantiate()'","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"once from the DispersiveShallowWater.jl main directory to tell Documenter.jl to build the documentation of your local clone. To build the documentation, run","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"julia --project=docs --color=yes docs/make.jl","category":"page"},{"location":"development/","page":"Development","title":"Development","text":"The resulting .html files can then be found in docs/build/ and you can look at them by opening them in a browser.","category":"page"},{"location":"overview/#Running-a-simulation","page":"Overview","title":"Running a simulation","text":"","category":"section"},{"location":"overview/#Introduction","page":"Overview","title":"Introduction","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"In this tutorial we describe how to numerically solve the BBM-BBM (Benjamin-Bona-Mahony) equations with variable bottom topography in one dimension, which has been proposed in [IsrawiKalischKatsaounisMitsotakis2021] for two spatial dimensions. The equations describe a dispersive shallow water model, i.e. they extend the well-known shallow water equations in the sense that dispersion is modeled. The shallow water equations are a system of first order hyperbolic partial differential equations that can be written in the form of a balance law. In contrast, the BBM-BBM equations additionally include third-order mixed derivatives. In primitive variables q = (eta v) they can be written as:","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"beginaligned\n eta_t + ((eta + D)v)_x - frac16(D^2eta_xt)_x = 0\n v_t + geta_x + left(frac12v^2right)_x - frac16(D^2v_t)_xx = 0\nendaligned","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"Here, eta = h + b describes the total water height, h the water height above the bottom topography (bathymetry), b = eta_0 - D the bathymetry and v the velocity in horizontal direction. Here, eta_0 is a reference water height also called still water height. In the case of the BBM-BBM equations, eta_0 is usually taken to be 0. The gravitational acceleration is denoted as g. A sketch of the water height and the bathymetry can be found below.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"(Image: water height and bathymetry)","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"In order to conduct a numerical simulation with DispersiveShallowWater.jl, we perform the following steps.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"First, we load the necessary libraries:","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"using DispersiveShallowWater, OrdinaryDiffEq","category":"page"},{"location":"overview/#Define-physical-setup","page":"Overview","title":"Define physical setup","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"As a first step of a numerical simulation, we define the physical setup we want to solve. This includes the set of equations, potentially including physical parameters, initial and boundary conditions as well as the domain. In the following example, the initial condition describes a travelling wave that moves towards a beach, which is modeled by a linearly increasing bathymetry.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"equations = BBMBBMVariableEquations1D(gravity_constant = 9.81)\n\nfunction initial_condition_shoaling(x, t, equations::BBMBBMVariableEquations1D, mesh)\n A = 0.07 # amplitude of wave\n x0 = -30 # initial center\n eta = A * exp(-0.1*(x - x0)^2)\n v = 0\n D = x <= 0.0 ? 0.7 : 0.7 - 1/50 * x\n return SVector(eta, v, D)\nend\n\ninitial_condition = initial_condition_shoaling\nboundary_conditions = boundary_condition_periodic\n\ncoordinates_min = -130.0\ncoordinates_max = 20.0\nN = 512\nmesh = Mesh1D(coordinates_min, coordinates_max, N + 1)","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"The first line specifies that we want to solve the BBM-BBM equations with variable bathymetry using a gravitational acceleration g = 981. Afterwards, we define the initial condition, which is described as a function with the spatial variable x, the time t, the equations and a mesh as parameters. If an analytical solution is available, the time variable t can be used, and the initial condition can serve as an analytical solution to be compared with the numerical solution. Otherwise, you can just keep the time variable unused. An initial condition in DispersiveShallowWater.jl is supposed to return an SVector holding the values for each of the unknown variables. Since the bathymetry is treated as a variable (with time derivative 0) for convenience, we need to provide the value for the primitive variables eta and v as well as for D.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"Next, we choose periodic boundary conditions since, up to now, DispersiveShallowWater.jl only provides support for this type of boundary conditions. Lastly, we define the physical domain as the interval from -130 to 20 and we choose 512 intermediate nodes. The mesh is homogeneous, i.e. the distance between each two nodes is constant. We choose the left boundary very far to the left in order to avoid an interaction of the left- and right-travelling waves.","category":"page"},{"location":"overview/#Define-numerical-solver","page":"Overview","title":"Define numerical solver","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"In the next step, we build a Semidiscretization that bundles all ingredients for the spatial discretization of the model. Especially, we need to define a Solver. The simplest way to define a solver is to call the constructor by providing the mesh and a desired order of accuracy. In the following example, we use an accuracy order of 4. The default constructor simply creates periodic first- and second-derivative central finite difference summation by parts operators of the provided order of accuracy. How to use other summation by parts operators, is described in the section on how to customize the solver.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"solver = Solver(mesh, 4)\n\nsemi = Semidiscretization(mesh, equations, initial_condition, solver, boundary_conditions = boundary_conditions)","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"Finally, we put the mesh, the equations, the initial_condition, the solver and the boundary_conditions together in a semidiscretization semi.","category":"page"},{"location":"overview/#Solve-system-of-ordinary-differential-equations","page":"Overview","title":"Solve system of ordinary differential equations","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"Once we have obtained a semidiscretization, we can solve the resulting system of ordinary differential equations. To do so, we specify the time interval that we want to simulate and obtain an ODEProblem from the SciML ecosystem for ordinary differential equations by calling semidiscretize on the semidiscretization and the time span. Additionally, we can analyze the numerical solution using an AnalysisCallback. The analysis includes computing the L^2 error and L^infty error of the different solution's variables compared to the initial condition (or, if available, at the same time analytical solution). Additional errors can be passed by the keyword argument extra_analysis_errors. Additional integral quantities that should be analyzed can be passed by keyword argument extra_analysis_integrals. In this example we pass the conservation_error, which computes the temporal change of the total amount (i.e. integral) of the different variables over time. In addition, the integrals of the total waterheight eta waterheight_total, the velocity and the entropy are computed and saved for each time step. The total waterheight and the total velocity are linear invariants of the BBM-BBM equations, i.e. they do not change over time. The total entropy","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"mathcal E(t eta v) = frac12int_Omega geta^2 + (eta + D)v^2textrmdx","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"is a nonlinear invariant and should be constant over time as well. During the simulation, the AnalysisCallback will print the results to the terminal.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"Finally, the ode can be solved using the interface from OrdinaryDiffEq.jl. This means, we can specify a time-stepping scheme we want to use the tolerances for the adaptive time-stepping and the time values, where the solution values should be saved. In this case, we use the adaptive explicit Runge-Kutta method Tsit5 by Tsitouras of order 5(4). Here, we save the solution at 100 equidistant points.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"tspan = (0.0, 25.0)\node = semidiscretize(semi, tspan)\nanalysis_callback = AnalysisCallback(semi; interval = 10,\n extra_analysis_errors = (:conservation_error,),\n extra_analysis_integrals = (waterheight_total,\n velocity, entropy),\n io = devnull)\ncallbacks = CallbackSet(analysis_callback)\n\nsaveat = range(tspan..., length = 100)\nsol = solve(ode, Tsit5(), abstol = 1e-7, reltol = 1e-7,\n save_everystep = false, callback = callbacks, saveat = saveat)","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"After solving the equations, sol contains the solution for each of the three variables at every spatial point for each of the 100 points in time. The errors and integrals recorded by the AnalysisCallback can be obtained as NamedTuples by errors(analysis_callback) and integrals(analysis_callback).","category":"page"},{"location":"overview/#visualize_results","page":"Overview","title":"Visualize results","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"After running the simulation, the results can be visualized using Plots.jl, which needs to be imported first. Then, we can plot the solution at the final time by calling plot on a Pair of the Semidiscretization and the corresponding ODESolution sol. The result is depicted in the following picture.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"using Plots\n\nplot(semi => sol)\nsavefig(\"shoaling_solution.png\") # hide\nnothing # hide","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"(Image: shoaling solution)","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"By default, this will plot the bathymetry, but not the initial (analytical) solution. You can adjust this by passing the boolean values plot_bathymetry (if true always plot to first subplot) and plot_initial. You can also provide a conversion function that converts the solution. A conversion function should take the values of the primitive variables q at one node, and the equations as input and should return an SVector of any length as output. For a user defined conversion function, there should also exist a function varnames(conversion, equations) that returns a Tuple of the variable names used for labelling. The conversion function can, e.g., be prim2cons or waterheight_total if one only wants to plot the total waterheight. The resulting plot will have one subplot for each of the returned variables of the conversion variable. By default, the conversion function is just prim2prim, i.e. the identity.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"Plotting an animation over time can, e.g., be done by the following command, which uses step to plot the solution at a specific time step.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"anim = @animate for step in 1:length(sol.u)\n plot(semi => sol, plot_initial = true, conversion = waterheight_total, step = step, xlim = (-50, 20), ylims = (-0.8, 0.1))\nend\ngif(anim, \"shoaling_solution.gif\", fps = 25)\nnothing # hide","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"(Image: shoaling solution)","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"It is also possible to plot the solution variables at a fixed spatial point over time by calling plot(semi => sol, x) for some x-value, see plot_examples.jl from the reproducibility repository of the master thesis of Joshua Lampert for some examples.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"Often, it is interesting to have a look at how the quantities that are recorded by the AnalysisCallback evolve in time. To this end, you can plot the AnalysisCallback by","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"plot(analysis_callback)\nsavefig(\"analysis_callback.png\") # hide\nnothing # hide","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"This creates the following figure:","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"(Image: analysis callback)","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"You can see that the linear invariants int_Omegaetatextrmdx and int_Omega vtextrmdx are indeed conserved exactly. The entropy, however, starts growing at around t = 17 and rises up to approximately 5e-5. This is because of the fact that, during the time integration, a nonlinear invariant is not necessarily conserved, even if the semidiscretization conserves the quantity exactly. How to obtain a fully-discrete structure-preserving numerical scheme is explained in the following section.","category":"page"},{"location":"overview/#Use-entropy-conserving-time-integration","page":"Overview","title":"Use entropy-conserving time integration","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"To obtain entropy-conserving time-stepping schemes DispersiveShallowWater.jl uses the relaxation method introduced in [Ketcheson2019] and further developed in [RanochaSayyariDalcinParsaniKetcheson2020]. The relaxation method is implemented as a RelaxationCallback, which takes a function representing the conserved quantity as the keyword argument invariant. Therefore, we can run the same example as above, but using relaxation on the entropy by simply adding another callback to the CallbackSet:","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"analysis_callback = AnalysisCallback(semi; interval = 10,\n extra_analysis_errors = (:conservation_error,),\n extra_analysis_integrals = (waterheight_total,\n velocity, entropy),\n io = devnull)\nrelaxation_callback = RelaxationCallback(invariant = entropy)\ncallbacks = CallbackSet(relaxation_callback, analysis_callback)\nsol = solve(ode, Tsit5(), abstol = 1e-7, reltol = 1e-7,\n save_everystep = false, callback = callbacks, saveat = saveat)","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"When you use both, an AnalysisCallback and a RelaxationCallback, note that the relaxation_callback needs to come first inside the CallbackSet as it needs to be invoked prior to the analysis_callback, such that the analysis_callback analyzes the solution with the already updated values.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"Plotting the analysis_callback again, we can see that now also the entropy is conserved up to machine precision.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"plot(analysis_callback, ylims = (-5e-16, 5e-16))\nsavefig(\"analysis_callback_relaxation.png\") # hide\nnothing # hide","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"(Image: analysis callback relaxation)","category":"page"},{"location":"overview/#customize_solver","page":"Overview","title":"Customize solver","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"In the semidiscretization created above, we used the default SBP operators, which are periodic finite difference operators. Using different SBP operators for the semidiscretization can be done leveraging SummationByPartsOperators.jl, which needs to be imported first:","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"using SummationByPartsOperators: legendre_derivative_operator, UniformPeriodicMesh1D, couple_discontinuously, PeriodicUpwindOperators","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"As an example, let us create a semidiscretization based on discontinuous Galerkin (DG) upwind operators. A semidiscretization implemented in DispersiveShallowWater.jl needs one first-derivative and one second-derivative SBP operator. To build the first-derivative operator, we first create a LegendreDerivativeOperator with polynomial degree 3 on a reference element [-1.0, 1.0] and a UniformPeriodicMesh1D for the coupling.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"mesh = Mesh1D(coordinates_min, coordinates_max, N)\naccuracy_order = 4\nD_legendre = legendre_derivative_operator(-1.0, 1.0, accuracy_order)\nuniform_mesh = UniformPeriodicMesh1D(mesh.xmin, mesh.xmax, div(mesh.N, accuracy_order))","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"Upwind DG operators in negative, central and positive operators can be obtained by couple_discontinuously","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"central = couple_discontinuously(D_legendre, uniform_mesh)\nminus = couple_discontinuously(D_legendre, uniform_mesh, Val(:minus))\nplus = couple_discontinuously(D_legendre, uniform_mesh, Val(:plus))\nD1 = PeriodicUpwindOperators(minus, central, plus)","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"In order to still have an entropy-conserving semidiscretization the second-derivative SBP operator needs to be","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"using SparseArrays: sparse\nD2 = sparse(plus) * sparse(minus)","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"The Solver object can now be created by passing the two SBP operators to the constructor, which, in turn, can be used to construct a Semidiscretization:","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"solver = Solver(D1, D2)\nsemi = Semidiscretization(mesh, equations, initial_condition, solver, boundary_conditions = boundary_conditions)","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"As before, we can run the simulation by","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"analysis_callback = AnalysisCallback(semi; interval = 10,\n extra_analysis_errors = (:conservation_error,),\n extra_analysis_integrals = (waterheight_total,\n velocity, entropy),\n io = devnull)\nrelaxation_callback = RelaxationCallback(invariant = entropy)\ncallbacks = CallbackSet(relaxation_callback, analysis_callback)\nsol = solve(ode, Tsit5(), abstol = 1e-7, reltol = 1e-7,\n save_everystep = false, callback = callbacks, saveat = saveat)\nanim = @animate for step in 1:length(sol.u)\n plot(semi => sol, plot_initial = true, conversion = waterheight_total, step = step, xlim = (-50, 20), ylims = (-0.8, 0.1))\nend\ngif(anim, \"shoaling_solution_dg.gif\", fps = 25)\nnothing # hide","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"(Image: shoaling solution DG)","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"For more details see also the documentation of SummationByPartsOperators.jl","category":"page"},{"location":"overview/#Additional-resources","page":"Overview","title":"Additional resources","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"Some more examples sorted by the simulated equations can be found in the examples/ subdirectory. Especially, in examples/svaerd_kalisch_1d/ you can find Julia scripts that solve the SvaerdKalischEquations1D that were not covered in this tutorial. The same steps as described above, however, apply in the same way to these equations. Attention must be paid for these equations because they do not conserve the classical total entropy mathcal E, but a modified entropy hatmathcal E, available as entropy_modified.","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"More examples, especially focussing on plotting, can be found in the scripts create_figures.jl and plot_examples.jl from the reproducibility repository of the master thesis of Joshua Lampert.","category":"page"},{"location":"overview/#References","page":"Overview","title":"References","text":"","category":"section"},{"location":"overview/","page":"Overview","title":"Overview","text":"[IsrawiKalischKatsaounisMitsotakis2021]: Israwi, Kalisch, Katsaounis, Mitsotakis (2021). A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis. DOI: 10.1088/1361-6544/ac3c29","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"[Ketcheson2019]: Ketcheson (2019): Relaxation Runge-Kutta Methods: Conservation and stability for Inner-Product Norms. DOI: 10.1137/19M1263662","category":"page"},{"location":"overview/","page":"Overview","title":"Overview","text":"[RanochaSayyariDalcinParsaniKetcheson2020]: Ranocha, Sayyari, Dalcin, Parsani, Ketcheson (2020): Relaxation Runge–Kutta Methods: Fully-Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier–Stokes Equations DOI: 10.1137/19M1263480","category":"page"},{"location":"ref/#DispersiveShallowWater.jl-API","page":"DispersiveShallowWater","title":"DispersiveShallowWater.jl API","text":"","category":"section"},{"location":"ref/","page":"DispersiveShallowWater","title":"DispersiveShallowWater","text":"CurrentModule = DispersiveShallowWater","category":"page"},{"location":"ref/","page":"DispersiveShallowWater","title":"DispersiveShallowWater","text":"Modules = [DispersiveShallowWater]","category":"page"},{"location":"ref/#DispersiveShallowWater.boundary_condition_periodic","page":"DispersiveShallowWater","title":"DispersiveShallowWater.boundary_condition_periodic","text":"boundary_condition_periodic = DispersiveShallowWater.BoundaryConditionPeriodic()\n\nA singleton struct indicating periodic boundary conditions.\n\n\n\n\n\n","category":"constant"},{"location":"ref/#DispersiveShallowWater.AbstractEquations","page":"DispersiveShallowWater","title":"DispersiveShallowWater.AbstractEquations","text":"AbstractEquations{NDIMS, NVARS}\n\nAn abstract supertype of specific equations such as the BBM-BBM equations. The type parameters encode the number of spatial dimensions (NDIMS) and the number of primary variables (NVARS) of the physics model.\n\n\n\n\n\n","category":"type"},{"location":"ref/#DispersiveShallowWater.AbstractSolver","page":"DispersiveShallowWater","title":"DispersiveShallowWater.AbstractSolver","text":"AbstractSolver\n\nAn abstract supertype of specific solvers.\n\n\n\n\n\n","category":"type"},{"location":"ref/#DispersiveShallowWater.AnalysisCallback","page":"DispersiveShallowWater","title":"DispersiveShallowWater.AnalysisCallback","text":"AnalysisCallback(semi; interval=0,\n extra_analysis_errors=Symbol[],\n extra_analysis_integrals=(),\n io=stdout)\n\nAnalyze a numerical solution every interval time steps. The L2- and the L∞-norm for each component are computed by default. Additional errors can be computed, e.g. by passing extra_analysis_errors = (:conservation_error,).\n\nFurther scalar functions func in extra_analysis_integrals are applied to the numerical solution and integrated over the computational domain. Some examples for this are entropy, and energy_total. You can also write your own function with the same signature as the examples listed above and pass it via extra_analysis_integrals. The computed errors and intergrals are saved for each timestep and can be obtained by calling errors and integrals.\n\nDuring the Simulation, the AnalysisCallback will print information to io.\n\n\n\n\n\n","category":"type"},{"location":"ref/#DispersiveShallowWater.BBMBBMEquations1D","page":"DispersiveShallowWater","title":"DispersiveShallowWater.BBMBBMEquations1D","text":"BBMBBMEquations1D(gravity, D, eta0 = 0.0)\n\nBBM-BBM (Benjamin–Bona–Mahony) system in one spatial dimension. The equations are given by\n\nbeginaligned\n eta_t + ((eta + D)v)_x - frac16D^2eta_xxt = 0\n v_t + geta_x + left(frac12v^2right)_x - frac16D^2v_xxt = 0\nendaligned\n\nThe unknown quantities of the BBM-BBM equations are the total water height eta and the velocity v. The gravitational constant is denoted by g and the constant bottom topography (bathymetry) b = eta_0 - D. The water height above the bathymetry is therefore given by h = eta - eta_0 + D. The BBM-BBM equations are only implemented for eta_0 = 0.\n\nOne reference for the BBM-BBM system can be found in\n\nJerry L. Bona, Min Chen (1998) A Boussinesq system for two-way propagation of nonlinear dispersive waves DOI: 10.1016/S0167-2789(97)00249-2\n\n\n\n\n\n","category":"type"},{"location":"ref/#DispersiveShallowWater.BBMBBMVariableEquations1D","page":"DispersiveShallowWater","title":"DispersiveShallowWater.BBMBBMVariableEquations1D","text":"BBMBBMVariableEquations1D(gravity, eta0 = 1.0)\n\nBBM-BBM (Benjamin–Bona–Mahony) system in one spatial dimension with spatially varying bathymetry. The equations are given by\n\nbeginaligned\n eta_t + ((eta + D)v)_x - frac16(D^2eta_xt)_x = 0\n v_t + geta_x + left(frac12v^2right)_x - frac16(D^2v_t)_xx = 0\nendaligned\n\nThe unknown quantities of the BBM-BBM equations are the total water height eta and the velocity v. The gravitational constant is denoted by g and the bottom topography (bathymetry) b = eta_0 - D. The water height above the bathymetry is therefore given by h = eta - eta_0 + D. The BBM-BBM equations are only implemented for eta_0 = 0.\n\nOne reference for the BBM-BBM system with spatially varying bathymetry can be found in\n\nSamer Israwi, Henrik Kalisch, Theodoros Katsaounis, Dimitrios Mitsotakis (2022) A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis DOI: 10.1088/1361-6544/ac3c29\n\n\n\n\n\n","category":"type"},{"location":"ref/#DispersiveShallowWater.Mesh1D","page":"DispersiveShallowWater","title":"DispersiveShallowWater.Mesh1D","text":"Mesh1D\n\nStruct that holds the information for a simple homogeneous one-dimensional mesh.\n\n\n\n\n\n","category":"type"},{"location":"ref/#DispersiveShallowWater.Mesh1D-Tuple{Any, Any, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.Mesh1D","text":"Mesh1D(xmin, xmax, N)\n\nCreate a simple homogeneous one-dimensional mesh from xmin to xmax with N nodes.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.RelaxationCallback","page":"DispersiveShallowWater","title":"DispersiveShallowWater.RelaxationCallback","text":"RelaxationCallback(invariant)\n\nUse a relaxation method in time in order to exactly preserve the (nonlinear) invariant for a conservative semidiscretization. A possible choice for invariant is invariant = entropy.\n\nReference\n\nHendrik Ranocha, Mohammed Sayyari, Lisandro Dalcin, Matteo Parsani, David I. Ketcheson (2020) Relaxation Runge–Kutta Methods: Fully-Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier–Stokes Equations DOI: 10.1137/19M1263480\n\n\n\n\n\n","category":"type"},{"location":"ref/#DispersiveShallowWater.Semidiscretization","page":"DispersiveShallowWater","title":"DispersiveShallowWater.Semidiscretization","text":"Semidiscretization\n\nA struct containing everything needed to describe a spatial semidiscretization of an equation.\n\n\n\n\n\n","category":"type"},{"location":"ref/#DispersiveShallowWater.Semidiscretization-NTuple{4, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.Semidiscretization","text":"Semidiscretization(mesh, equations, initial_condition, solver;\n source_terms=nothing,\n boundary_conditions=boundary_condition_periodic,\n RealT=real(solver),\n uEltype=RealT,\n initial_cache=NamedTuple())\n\nConstruct a semidiscretization of a PDE.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.Solver","page":"DispersiveShallowWater","title":"DispersiveShallowWater.Solver","text":"Solver\n\nA struct that holds the summation by parts (SBP) operators that are used for the spatial discretization.\n\n\n\n\n\n","category":"type"},{"location":"ref/#DispersiveShallowWater.Solver-Tuple{Any, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.Solver","text":"Solver(mesh, accuracy_order)\n\nCreate a solver, where the summation by parts (SBP) operators are of order accuracy_order and associated to the mesh.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.Solver-Union{Tuple{RealT}, Tuple{SummationByPartsOperators.AbstractDerivativeOperator{RealT}, Union{AbstractMatrix{RealT}, SummationByPartsOperators.AbstractDerivativeOperator{RealT}}}} where RealT","page":"DispersiveShallowWater","title":"DispersiveShallowWater.Solver","text":"Solver(D1, D2)\n\nCreate a solver, where D1 is an AbstractDerivativeOperator from SummationByPartsOperators.jl of first derivative_order and D2 is an AbstractDerivativeOperator of second derivative_order or an AbstractMatrix. Both summation by parts operators should be associated with the same grid.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.SummaryCallback","page":"DispersiveShallowWater","title":"DispersiveShallowWater.SummaryCallback","text":"SummaryCallback(io::IO = stdout)\n\nCreate and return a callback that resets the timer at the beginning of a simulation and prints the timer values at the end of the simulation.\n\n\n\n\n\n","category":"type"},{"location":"ref/#DispersiveShallowWater.SvaerdKalischEquations1D","page":"DispersiveShallowWater","title":"DispersiveShallowWater.SvaerdKalischEquations1D","text":"SvärdKalischEquations1D(gravity, eta0 = 1.0, alpha = 0.0, beta = 0.2308939393939394, gamma = 0.04034343434343434)\n\nDispersive system by Svärd and Kalisch in one spatial dimension with spatially varying bathymetry. The equations are given in conservative variables by\n\nbeginaligned\n h_t + (hv)_x = (hatalpha(hatalpha(h + b)_x)_x)_x\n (hv)_t + (hv^2)_x + gh(h + b)_x = (hatalpha v(hatalpha(h + b)_x)_x)_x + (hatbeta v_x)_xt + frac12(hatgamma v_x)_xx + frac12(hatgamma v_xx)_x\nendaligned\n\nwhere hatalpha^2 = alphasqrtgDD^2, hatbeta = beta D^3, hatgamma = gammasqrtgDD^3. The coefficients alpha, beta and gamma are provided in dimensionless form and D = eta_0 - b is the still-water depth and eta0 is the still-water surface (lake-at-rest). The equations can be rewritten in primitive variables as\n\nbeginaligned\n eta_t + ((eta + D)v)_x = (hatalpha(hatalphaeta_x)_x)_x\n v_t(eta + D) - v((eta + D)v)_x + ((eta + D)v^2)_x + g(eta + D)eta_x = (hatalpha v(hatalphaeta_x)_x)_x - v(hatalpha(hatalphaeta_x)_x)_x + (hatbeta v_x)_xt + frac12(hatgamma v_x)_xx + frac12(hatgamma v_xx)_x\nendaligned\n\nThe unknown quantities of the Svärd-Kalisch equations are the total water height eta and the velocity v. The gravitational constant is denoted by g and the bottom topography (bathymetry) b = eta_0 - D. The water height above the bathymetry is therefore given by h = eta - eta_0 + D.\n\nThe equations by Svärd and Kalisch are presented and analyzed in\n\nMagnus Svärd, Henrik Kalisch (2023) A novel energy-bounded Boussinesq model and a well-balanced and stable numerical discretization arXiv: 2302.09924\n\n\n\n\n\n","category":"type"},{"location":"ref/#DispersiveShallowWater.cons2prim","page":"DispersiveShallowWater","title":"DispersiveShallowWater.cons2prim","text":"cons2prim(u, equations)\n\nConvert the conserved variables u to the primitive variables for a given set of equations. u is a vector type of the correct length nvariables(equations). Notice the function doesn't include any error checks for the purpose of efficiency, so please make sure your input is correct. The inverse conversion is performed by prim2cons.\n\n\n\n\n\n","category":"function"},{"location":"ref/#DispersiveShallowWater.convergence_test-Tuple{Module, AbstractString, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.convergence_test","text":"convergence_test([mod::Module=Main,] example::AbstractString, iterations; kwargs...)\n\nRun iterations simulations using the setup given in example and compute the experimental order of convergence (EOC) in the L^2 and L^infty norm. In each iteration, the resolution of the respective mesh will be doubled. Additional keyword arguments kwargs... and the optional module mod are passed directly to trixi_include.\n\nAdjusted from Trixi.jl.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.default_analysis_errors-Tuple{DispersiveShallowWater.AbstractEquations}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.default_analysis_errors","text":"default_analysis_errors(equations)\n\nDefault analysis errors used by the AnalysisCallback.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.default_analysis_integrals-Tuple{DispersiveShallowWater.AbstractEquations}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.default_analysis_integrals","text":"default_analysis_integrals(equations)\n\nDefault analysis integrals used by the AnalysisCallback.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.default_example-Tuple{}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.default_example","text":"default_example()\n\nReturn the path to an example that can be used to quickly see DispersiveShallowWater.jl in action. See also examples_dir and get_examples.\n\nCopied from Trixi.jl.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.discharge-Tuple{Any, DispersiveShallowWater.AbstractEquations}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.discharge","text":"discharge(q, equations)\n\nSee momentum.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.eachvariable-Tuple{DispersiveShallowWater.AbstractEquations}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.eachvariable","text":"eachvariable(equations::AbstractEquations)\n\nReturn an iterator over the indices that specify the location in relevant data structures for the variables in equations. In particular, not the variables themselves are returned.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.energy_total","page":"DispersiveShallowWater","title":"DispersiveShallowWater.energy_total","text":"energy_total(q, equations)\n\nReturn the total energy of the primitive variables q for a given set of equations.\n\nq is a vector of the primitive variables at a single node, i.e., a vector of the correct length nvariables(equations).\n\n\n\n\n\n","category":"function"},{"location":"ref/#DispersiveShallowWater.energy_total_modified-Tuple{Any, SvaerdKalischEquations1D, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.energy_total_modified","text":"energy_total_modified(q, equations::SvaerdKalischEquations1D, cache)\n\nReturn the modified total energy of the primitive variables q for the SvaerdKalischEquations1D. It contains an additional term containing a derivative compared to the usual energy_total. The energy_total_modified is a conserved quantity of the Svärd-Kalisch equations.\n\nq is a vector of the primitive variables at ALL nodes, i.e., a matrix of the correct length nvariables(equations) as first dimension and the number of nodes as length of the second dimension. cache needs to hold the first-derivative SBP operator D1.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.entropy","page":"DispersiveShallowWater","title":"DispersiveShallowWater.entropy","text":"entropy(q, equations)\n\nReturn the entropy of the primitive variables q for a given set of equations.\n\nq is a vector of the primitive variables at a single node, i.e., a vector of the correct length nvariables(equations).\n\n\n\n\n\n","category":"function"},{"location":"ref/#DispersiveShallowWater.entropy_modified-Tuple{Any, SvaerdKalischEquations1D, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.entropy_modified","text":"entropy_modified(q, equations::SvaerdKalischEquations1D, cache)\n\nAlias for energy_total_modified.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.errors-Union{Tuple{SciMLBase.DiscreteCallback{Condition, Affect!}}, Tuple{Affect!}, Tuple{Condition}} where {Condition, Affect!<:AnalysisCallback}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.errors","text":"errors(analysis_callback)\n\nReturn the computed errors for each timestep as a named tuple. The shape of each entry is (nvariables, ntimesteps).\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.examples_dir-Tuple{}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.examples_dir","text":"examples_dir()\n\nReturn the directory where the example files provided with DispersiveShallowWater.jl are located. If DispersiveShallowWater is installed as a regular package (with ]add DispersiveShallowWater), these files are read-only and should not be modified. To find out which files are available, use, e.g., readdir.\n\nCopied from Trixi.jl.\n\nExamples\n\nreaddir(examples_dir())\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.get_examples-Tuple{}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.get_examples","text":"get_examples()\n\nReturn a list of all examples that are provided by DispersiveShallowWater.jl. See also examples_dir and default_example.\n\nCopied from Trixi.jl.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.get_name-Tuple{DispersiveShallowWater.AbstractEquations}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.get_name","text":"get_name(equations::AbstractEquations)\n\nReturn the canonical, human-readable name for the given system of equations.\n\nExamples\n\njulia> DispersiveShallowWater.get_name(BBMBBMEquations1D(gravity_constant=1.0))\n\"BBMBBMEquations1D\"\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.initial_condition_convergence_test-Tuple{Any, Any, BBMBBMEquations1D, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.initial_condition_convergence_test","text":"initial_condition_convergence_test(x, t, equations::BBMBBMEquations1D, mesh)\n\nA travelling-wave solution used for convergence tests in a periodic domain.\n\nFor details see Example 5 in Section 3 from (here adapted for dimensional equations):\n\nMin Chen (1997) Exact Traveling-Wave Solutions to Bidirectional Wave Equations DOI: 10.1023/A:1026667903256\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.initial_condition_convergence_test-Tuple{Any, Any, BBMBBMVariableEquations1D, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.initial_condition_convergence_test","text":"initial_condition_convergence_test(x, t, equations::BBMBBMVariableEquations1D, mesh)\n\nA travelling-wave solution used for convergence tests in a periodic domain. The bathymetry is constant.\n\nFor details see Example 5 in Section 3 from (here adapted for dimensional equations):\n\nMin Chen (1997) Exact Traveling-Wave Solutions to Bidirectional Wave Equations DOI: 10.1023/A:1026667903256\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.initial_condition_dingemans-Tuple{Any, Any, BBMBBMVariableEquations1D, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.initial_condition_dingemans","text":"initial_condition_dingemans(x, t, equations::BBMBBMVariableEquations1D, mesh)\n\nThe initial condition that uses the dispersion relation of the Euler equations to approximate waves generated by a wave maker as it is done by experiments of Dingemans. The topography is a trapezoidal.\n\nwarning: Translation of water height\nThe initial condition for the water height is translated to be around 0, which is needed for the simulation because the BBMBBMVariableEquations1D are only implemented for ta_0 = 0.\n\nReferences:\n\nMagnus Svärd, Henrik Kalisch (2023) A novel energy-bounded Boussinesq model and a well-balanced and stable numerical discretization arXiv: 2302.09924\nMaarten W. Dingemans (1994) Comparison of computations with Boussinesq-like models and laboratory measurements link\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.initial_condition_dingemans-Tuple{Any, Any, SvaerdKalischEquations1D, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.initial_condition_dingemans","text":"initial_condition_dingemans(x, t, equations::SvaerdKalischEquations1D, mesh)\n\nThe initial condition that uses the dispersion relation of the Euler equations to approximate waves generated by a wave maker as it is done by experiments of Dingemans. The topography is a trapezoidal. It is assumed that equations.eta0 = 0.8.\n\nReferences:\n\nMagnus Svärd, Henrik Kalisch (2023) A novel energy-bounded Boussinesq model and a well-balanced and stable numerical discretization arXiv: 2302.09924\nMaarten W. Dingemans (1994) Comparison of computations with Boussinesq-like models and laboratory measurements link\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.initial_condition_manufactured-Tuple{Any, Any, BBMBBMEquations1D, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.initial_condition_manufactured","text":"initial_condition_manufactured(x, t, equations::BBMBBMEquations1D, mesh)\n\nA smooth manufactured solution in combination with source_terms_manufactured.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.initial_condition_manufactured-Tuple{Any, Any, BBMBBMVariableEquations1D, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.initial_condition_manufactured","text":"initial_condition_manufactured(x, t, equations::BBMBBMVariableEquations1D, mesh)\n\nA smooth manufactured solution in combination with source_terms_manufactured.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.initial_condition_manufactured-Tuple{Any, Any, SvaerdKalischEquations1D, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.initial_condition_manufactured","text":"initial_condition_manufactured(x, t, equations::SvaerdKalischEquations1D, mesh)\n\nA smooth manufactured solution in combination with source_terms_manufactured.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.integrals-Union{Tuple{SciMLBase.DiscreteCallback{Condition, Affect!}}, Tuple{Affect!}, Tuple{Condition}} where {Condition, Affect!<:AnalysisCallback}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.integrals","text":"integrals(analysis_callback)\n\nReturn the computed integrals for each timestep as a named tuple.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.momentum-Tuple{Any, DispersiveShallowWater.AbstractEquations}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.momentum","text":"momentum(q, equations)\n\nReturn the momentum of the primitive variables q for a given set of equations, i.e. the waterheight times the velocity.\n\nq is a vector of the primitive variables at a single node, i.e., a vector of the correct length nvariables(equations).\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.prim2cons","page":"DispersiveShallowWater","title":"DispersiveShallowWater.prim2cons","text":"prim2cons(q, equations)\n\nConvert the primitive variables q to the conserved variables for a given set of equations. q is a vector type of the correct length nvariables(equations). Notice the function doesn't include any error checks for the purpose of efficiency, so please make sure your input is correct. The inverse conversion is performed by cons2prim.\n\n\n\n\n\n","category":"function"},{"location":"ref/#DispersiveShallowWater.prim2prim-Tuple{Any, DispersiveShallowWater.AbstractEquations}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.prim2prim","text":"prim2prim(q, equations)\n\nReturn the primitive variables q. While this function is as trivial as identity, it is also as useful.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.semidiscretize-Tuple{Semidiscretization, Any}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.semidiscretize","text":"semidiscretize(semi::Semidiscretization, tspan)\n\nWrap the semidiscretization semi as an ODE problem in the time interval tspan that can be passed to solve from the SciML ecosystem.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.source_terms_manufactured-Tuple{Any, Any, Any, BBMBBMEquations1D}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.source_terms_manufactured","text":"source_terms_manufactured(q, x, t, equations::BBMBBMEquations1D, mesh)\n\nA smooth manufactured solution in combination with initial_condition_manufactured.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.source_terms_manufactured-Tuple{Any, Any, Any, BBMBBMVariableEquations1D}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.source_terms_manufactured","text":"source_terms_manufactured(q, x, t, equations::BBMBBMVariableEquations1D, mesh)\n\nA smooth manufactured solution in combination with initial_condition_manufactured.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.source_terms_manufactured-Tuple{Any, Any, Any, SvaerdKalischEquations1D}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.source_terms_manufactured","text":"source_terms_manufactured(q, x, t, equations::SvaerdKalischEquations1D, mesh)\n\nA smooth manufactured solution in combination with initial_condition_manufactured.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.tstops-Union{Tuple{SciMLBase.DiscreteCallback{Condition, Affect!}}, Tuple{Affect!}, Tuple{Condition}} where {Condition, Affect!<:AnalysisCallback}","page":"DispersiveShallowWater","title":"DispersiveShallowWater.tstops","text":"tstops(analysis_callback)\n\nReturn the time values that correspond to the saved values of the errors and integrals.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.varnames","page":"DispersiveShallowWater","title":"DispersiveShallowWater.varnames","text":"varnames(conversion_function, equations)\n\nReturn the list of variable names when applying conversion_function to the conserved variables associated to equations. Common choices of the conversion_function are prim2prim and prim2cons.\n\n\n\n\n\n","category":"function"},{"location":"ref/#DispersiveShallowWater.velocity","page":"DispersiveShallowWater","title":"DispersiveShallowWater.velocity","text":"velocity(q, equations)\n\nReturn the velocity of the primitive variables q for a given set of equations.\n\nq is a vector of the primitive variables at a single node, i.e., a vector of the correct length nvariables(equations).\n\n\n\n\n\n","category":"function"},{"location":"ref/#DispersiveShallowWater.waterheight","page":"DispersiveShallowWater","title":"DispersiveShallowWater.waterheight","text":"waterheight(q, equations)\n\nReturn the waterheight of the primitive variables q for a given set of equations, i.e. the waterheight above the bathymetry.\n\nq is a vector of the primitive variables at a single node, i.e., a vector of the correct length nvariables(equations).\n\n\n\n\n\n","category":"function"},{"location":"ref/#DispersiveShallowWater.waterheight_total","page":"DispersiveShallowWater","title":"DispersiveShallowWater.waterheight_total","text":"waterheight_total(q, equations)\n\nReturn the total waterheight of the primitive variables q for a given set of equations, i.e. the waterheight plus the bathymetry.\n\nq is a vector of the primitive variables at a single node, i.e., a vector of the correct length nvariables(equations).\n\n\n\n\n\n","category":"function"},{"location":"ref/#SummationByPartsOperators.grid-Tuple{Semidiscretization}","page":"DispersiveShallowWater","title":"SummationByPartsOperators.grid","text":"grid(semi)\n\nGet the grid of a semidiscretization.\n\n\n\n\n\n","category":"method"},{"location":"ref/#DispersiveShallowWater.@autoinfiltrate","page":"DispersiveShallowWater","title":"DispersiveShallowWater.@autoinfiltrate","text":"@autoinfiltrate\n@autoinfiltrate condition::Bool\n\nInvoke the @infiltrate macro of the package Infiltrator.jl to create a breakpoint for ad-hoc interactive debugging in the REPL. If the optional argument condition is given, the breakpoint is only enabled if condition evaluates to true.\n\nAs opposed to using Infiltrator.@infiltrate directly, this macro does not require Infiltrator.jl to be added as a dependency to DispersiveShallowWater.jl. As a bonus, the macro will also attempt to load the Infiltrator module if it has not yet been loaded manually.\n\nNote: For this macro to work, the Infiltrator.jl package needs to be installed in your current Julia environment stack.\n\nSee also: Infiltrator.jl\n\nwarning: Internal use only\nPlease note that this macro is intended for internal use only. It is not part of the public API of DispersiveShallowWater.jl, and it thus can altered (or be removed) at any time without it being considered a breaking change.\n\n\n\n\n\n","category":"macro"},{"location":"#DispersiveShallowWater.jl","page":"Home","title":"DispersiveShallowWater.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"(Image: Docs-stable) (Image: Docs-dev) (Image: Build Status) (Image: codecov) (Image: Coveralls) (Image: Aqua QA) (Image: License: MIT) (Image: DOI)","category":"page"},{"location":"","page":"Home","title":"Home","text":"DispersiveShallowWater.jl is a Julia package that implements structure-preserving numerical methods for dispersive shallow water models. To date, it provides provably conservative, entropy-conserving and well-balanced numerical schemes for two dispersive shallow water models:","category":"page"},{"location":"","page":"Home","title":"Home","text":"the BBM-BBM equations with varying bottom topography,\nthe dispersive shallow water model proposed by Magnus Svärd and Henrik Kalisch.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The semidiscretizations are based on summation by parts (SBP) operators, which are implemented in SummationByPartsOperators.jl. In order to obtain fully discrete schemes, the time integration methods from OrdinaryDiffEq.jl are used to solve the resulting ordinary differential equations. Fully discrete entropy-conservative methods can be obtained by using the relaxation method provided by DispersiveShallowWater.jl.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"If you have not yet installed Julia, then you first need to download Julia. Please follow the instructions for your operating system. DispersiveShallowWater.jl works with Julia v1.9 and newer. DispersiveShallowWater.jl is a registered Julia package. Therefore, you can install it by executing the following commands from the Julia REPL","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> using Pkg\n\njulia> Pkg.add([\"DispersiveShallowWater\", \"OrdinaryDiffEq\", \"Plots\"])","category":"page"},{"location":"","page":"Home","title":"Home","text":"In addition, this installs the packages OrdinaryDiffEq.jl used for time-integration and Plots.jl to visualize the results. If you want to use other SBP operators than the default operators that DispersiveShallowWater.jl uses, then you also need SummationByPartsOperators.jl, which can be installed running","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> Pkg.add(\"SummationByPartsOperators\")","category":"page"},{"location":"#Usage","page":"Home","title":"Usage","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"In the Julia REPL, first load the package DispersiveShallowWater.jl","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> using DispersiveShallowWater","category":"page"},{"location":"","page":"Home","title":"Home","text":"You can run a basic simulation that solves the BBM-BBM equations by executing","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> include(default_example());","category":"page"},{"location":"","page":"Home","title":"Home","text":"The result can be visualized by using the package Plots.jl","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> using Plots\njulia> plot(semi => sol)","category":"page"},{"location":"","page":"Home","title":"Home","text":"The command plot expects a Pair consisting of a Semidiscretization and an ODESolution. The visualization can also be customized, see the documentation for more details. Other examples can be found in the subdirectory examples/. A list of all examples is returned by running get_examples(). You can pass the filename of one of the examples or your own simulation file to include in order to run it, e.g., include(joinpath(examples_dir(), \"svaerd_kalisch_1d\", \"svaerd_kalisch_1d_dingemans_relaxation.jl\")).","category":"page"},{"location":"#Referencing","page":"Home","title":"Referencing","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"You can directly refer to DispersiveShallowWater.jl as","category":"page"},{"location":"","page":"Home","title":"Home","text":"@misc{lampert2023dispersive,\n title={{D}ispersive{S}hallow{W}ater.jl: {S}tructure-preserving numerical\n methods for dispersive shallow water models},\n author={Lampert, Joshua},\n year={2023},\n month={10},\n howpublished={\\url{https://github.com/JoshuaLampert/DispersiveShallowWater.jl}},\n doi={10.5281/zenodo.10034636}\n}","category":"page"},{"location":"#Authors","page":"Home","title":"Authors","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"The package is developed and maintained by Joshua Lampert (University of Hamburg). Some parts of this repository are based on parts of Dispersive-wave-schemes-notebooks. A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations by Hendrik Ranocha, Dimitrios Mitsotakis and David Ketcheson. The code structure is inspired by Trixi.jl.","category":"page"},{"location":"#License-and-contributing","page":"Home","title":"License and contributing","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"DispersiveShallowWater.jl is published under the MIT license (see License). We are pleased to accept contributions from everyone, preferably in the form of a PR.","category":"page"}] +} diff --git a/v0.3.2/shoaling_solution.gif b/v0.3.2/shoaling_solution.gif new file mode 100644 index 0000000000000000000000000000000000000000..b1ae9da870f797a35e892eb30bab07ba858d1be5 GIT binary patch literal 555785 zcmeF(S5y;U_&54VAqhPc>0qeRJ4z=B9RUGR5CMarfC4|HgMbNLYCuI)(9k;qf&z+$ z-ULDw5Ftpf(z|l_zwgaiXRUK_Zr^o2YhTRW%$l{nGkZU?pD{ExP*roX1rtFBAmD$T z0U!_v0)arGP&gb;OH0eZz`!V~#B`2y%xw4&nSIQ1F1njF`-q*Lor8me>s;P34-XG- z{xLs4|Hb@6K|w*0bHyj(;^IEPgi!{MBro$uYdcmMu{UjFRaRD3 zP3%^O`&XZvI<7tU`|rmmejh(IeIf+Z)zy7|k@ET6+`qa%jM~|9?pSPYlWbTSESijmQ6&Y@EmAe?y!BN3?uK zm87Of1e>INXJvLvEK=Ac)2J$^ErCnv!DMGuZbu4AH=fV9I8p^t$jiRQ9fwzF=9Qx||r`ZIS~8UUB6BfZ=Z; zQKInGjpx=hcxAc8EcKmXN|WdPeD8tWgkmFiD{F7uqqfBsQ*EOl3QK2=XS5@*G*$i? z{X7@qv?cJiWsi(#;4Mdy?q933&y7Mp=TWIDln=`%EMdeFa8%<@e0w z5JHYijHEZz%nX|O-OlpPZQ5aBz81#Fb%l5@qjnby`Xcd&qHqGL1abcMe2#&vDu^)R zmx|cwe7@YmCFov5$n!BDOx_p5)T`jyr3Bu^2Ca 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