Adapt Svärd-Kalisch equations to Serre-Green-Naghdi equations (#146)

* adapt Svärd-Kalisch equations to style used in Serre-Green-Nagdhi equations * simplify Serre-Green-Naghdi a bit * remove allocations in modified entropy * simplify modified entropy * reduce allocations * format * simplify * Apply suggestions from code review Co-authored-by: Hendrik Ranocha <[email protected]> * render in math font * clarify that only variable bathymetry is supported * introduce mutating and non-mutating energy_total_modified * format * simplify integrate_quantity * add tests for energy_total_modified and entropy_modified * call mutating entropy_modified inside entropy_modified * document mutating version of energy_total_modified * Apply suggestions from code review Co-authored-by: Hendrik Ranocha <[email protected]> * use inplace version of function * Update src/equations/serre_green_naghdi_1d.jl --------- Co-authored-by: Hendrik Ranocha <[email protected]>
JoshuaLampert · Aug 25, 2024 · 1f8f932 · 1f8f932
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diff --git a/src/callbacks_step/analysis.jl b/src/callbacks_step/analysis.jl
@@ -298,7 +298,7 @@ function analyze_integrals!(io, current_integrals, i, analysis_integrals::NTuple
     quantity = first(analysis_integrals)
     remaining_quantities = Base.tail(analysis_integrals)
 
-    res = analyze(quantity, q, t, semi)
+    res = analyze!(semi, quantity, q, t)
     current_integrals[i] = res
     @printf(io, " %-12s:", pretty_form_utf(quantity))
     @printf(io, "  % 10.8e", res)
@@ -326,17 +326,8 @@ function (cb::DiscreteCallback{Condition, Affect!})(sol) where {Condition,
     return (; l2 = l2_error, linf = linf_error)
 end
 
-function analyze(quantity, q, t, semi::Semidiscretization)
-    integrate_quantity(q -> quantity(q, semi.equations), q, semi)
-end
-
-# The modified entropy/energy of the Svärd-Kalisch and
-# Serre-Green-Naghdi equations
-# takes the whole `q` for every point in space since it requires
-# the derivative of the velocity `v_x`.
-function analyze(quantity::Union{typeof(energy_total_modified), typeof(entropy_modified)},
-                 q, t, semi::Semidiscretization)
-    integrate_quantity(quantity, q, semi)
+function analyze!(semi::Semidiscretization, quantity, q, t)
+    integrate_quantity!(semi.cache.tmp1, quantity, q, semi)
 end
 
 pretty_form_utf(::typeof(waterheight_total)) = "∫η"

diff --git a/src/callbacks_step/relaxation.jl b/src/callbacks_step/relaxation.jl
@@ -69,15 +69,7 @@ end
 
     function relaxation_functional(q, semi)
         @unpack tmp1 = semi.cache
-        # modified entropy from Svärd-Kalisch equations need to take the whole vector `q` for every point in space
-        if relaxation_callback.invariant isa
-           Union{typeof(energy_total_modified), typeof(entropy_modified)}
-            return integrate_quantity!(tmp1, relaxation_callback.invariant, q, semi)
-        else
-            return integrate_quantity!(tmp1,
-                                       q -> relaxation_callback.invariant(q, semi.equations),
-                                       q, semi)
-        end
+        return integrate_quantity!(tmp1, relaxation_callback.invariant, q, semi)
     end
 
     function convex_combination(gamma, old, new)

diff --git a/src/equations/equations.jl b/src/equations/equations.jl
@@ -189,7 +189,7 @@ of the correct length `nvariables(equations)`.
 function bathymetry end
 
 """
-        lake_at_rest_error(q, equations::AbstractShallowWaterEquations)
+    lake_at_rest_error(q, equations::AbstractShallowWaterEquations)
 
 Calculate the error for the "lake-at-rest" test case where the
 [`waterheight_total`](@ref) ``\\eta = h + b`` should
@@ -270,12 +270,25 @@ contributions.
 `q_global` is a vector of the primitive variables at ALL nodes.
 `cache` needs to hold the SBP operators used by the `solver` if non-hydrostatic
 terms are present.
+
+Internally, this function allocates a vector for the output and
+calls [`DispersiveShallowWater.energy_total_modified!`](@ref).
 """
 function energy_total_modified(q_global, equations::AbstractShallowWaterEquations, cache)
+    e = similar(q_global.x[begin])
+    return energy_total_modified!(e, q_global, equations, cache)
+end
+
+"""
+    energy_total_modified!(e, q_global, equations::AbstractShallowWaterEquations, cache)
+
+In-place version of [`energy_total_modified`](@ref).
+"""
+function energy_total_modified!(e, q_global, equations::AbstractShallowWaterEquations,
+                                cache)
     # `q_global` is an `ArrayPartition` of the primitive variables at all nodes
     @assert nvariables(equations) == length(q_global.x)
 
-    e = similar(q_global.x[begin])
     for i in eachindex(q_global.x[begin])
         e[i] = energy_total(get_node_vars(q_global, equations, i), equations)
     end
@@ -291,9 +304,20 @@ varnames(::typeof(energy_total_modified), equations) = ("e_modified",)
 Alias for [`energy_total_modified`](@ref).
 """
 @inline function entropy_modified(q_global, equations::AbstractShallowWaterEquations, cache)
-    energy_total_modified(q_global, equations, cache)
+    e = similar(q_global.x[begin])
+    return entropy_modified!(e, q_global, equations, cache)
 end
 
+"""
+    entropy_modified!(e, q_global, equations::AbstractShallowWaterEquations, cache)
+
+In-place version of [`entropy_modified`](@ref).
+"""
+@inline entropy_modified!(e, q_global, equations, cache) = energy_total_modified!(e,
+                                                                                  q_global,
+                                                                                  equations,
+                                                                                  cache)
+
 varnames(::typeof(entropy_modified), equations) = ("U_modified",)
 
 # Add methods to show some information on systems of equations.
@@ -448,3 +472,27 @@ abstract type AbstractSerreGreenNaghdiEquations{NDIMS, NVARS} <:
 const AbstractSerreGreenNaghdiEquations1D = AbstractSerreGreenNaghdiEquations{1}
 include("serre_green_naghdi_1d.jl")
 include("hyperbolic_serre_green_naghdi_1d.jl")
+
+function solve_system_matrix!(dv, system_matrix, rhs,
+                              ::Union{SvaerdKalischEquations1D,
+                                      SerreGreenNaghdiEquations1D},
+                              D1, cache)
+    if issparse(system_matrix)
+        (; factorization) = cache
+        cholesky!(factorization, system_matrix; check = false)
+        if issuccess(factorization)
+            scale_by_mass_matrix!(rhs, D1)
+            # see https://github.com/JoshuaLampert/DispersiveShallowWater.jl/issues/122
+            dv .= factorization \ rhs
+        else
+            # The factorization may fail if the time step is too large
+            # and h becomes negative.
+            fill!(dv, NaN)
+        end
+    else
+        factorization = cholesky!(system_matrix)
+        scale_by_mass_matrix!(rhs, D1)
+        ldiv!(dv, factorization, rhs)
+    end
+    return nothing
+end
diff --git a/src/equations/hyperbolic_serre_green_naghdi_1d.jl b/src/equations/hyperbolic_serre_green_naghdi_1d.jl
@@ -433,9 +433,9 @@ end
 
 # The entropy/energy takes the whole `q` for every point in space
 """
-    energy_total_modified(q_global, equations::HyperbolicSerreGreenNaghdiEquations1D, cache)
+    DispersiveShallowWater.energy_total_modified!(e, q_global, equations::HyperbolicSerreGreenNaghdiEquations1D, cache)
 
-Return the modified total energy of the primitive variables `q_global` for the
+Return the modified total energy `e` of the primitive variables `q_global` for the
 [`HyperbolicSerreGreenNaghdiEquations1D`](@ref).
 It contains additional terms compared to the usual [`energy_total`](@ref)
 modeling non-hydrostatic contributions. The `energy_total_modified`
@@ -449,10 +449,12 @@ the total modified energy is given by
 ```
 
 `q_global` is a vector of the primitive variables at ALL nodes.
+
+See also [`energy_total_modified`](@ref).
 """
-function energy_total_modified(q_global,
-                               equations::HyperbolicSerreGreenNaghdiEquations1D,
-                               cache)
+function energy_total_modified!(e, q_global,
+                                equations::HyperbolicSerreGreenNaghdiEquations1D,
+                                cache)
     # unpack physical parameters and SBP operator `D1`
     g = gravity_constant(equations)
     (; lambda) = equations
@@ -464,8 +466,6 @@ function energy_total_modified(q_global,
     @.. b = equations.eta0 - D
     @.. h = eta - b
 
-    e = zero(h)
-
     # 1/2 g eta^2 + 1/2 h v^2 + 1/6 h^3 w^2 + λ/6 h (1 - H/h)^2
     @.. e = 1 / 2 * g * eta^2 + 1 / 2 * h * v^2 + 1 / 6 * h * w^2 +
             lambda / 6 * h * (1 - H / h)^2

diff --git a/src/equations/serre_green_naghdi_1d.jl b/src/equations/serre_green_naghdi_1d.jl
@@ -43,7 +43,7 @@ approximation, the Serre-Green-Naghdi equations are
     + \frac{1}{2} g (h^2)_x + g h b_x + \frac{1}{2} h (v^2)_x
     + p_x + \frac{3}{2} \frac{p}{h} b_x + \psi b_x &= 0,\\
   p &= \frac{1}{3} h^3 v_{x}^2 - \frac{1}{3} h^3 v v_{xx}
-    + \frac{1}{2} h^2 v (b_x v)_x,
+    + \frac{1}{2} h^2 v (b_x v)_x,\\
   \psi &= \frac{1}{4} h v (b_x v)_x.
 \end{aligned}
 ```
@@ -57,8 +57,8 @@ References for the Serre-Green-Naghdi system can be found in
   [DOI: 10.1017/S0022112076002425](https://doi.org/10.1017/S0022112076002425)
 
 The semidiscretization implemented here conserves
-- the total water mass (integral of h) as a linear invariant
-- the total momentum (integral of h v) as a nonlinear invariant if the bathymetry is constant
+- the total water mass (integral of ``h``) as a linear invariant
+- the total momentum (integral of ``h v``) as a nonlinear invariant if the bathymetry is constant
 - the total modified energy
 
 for periodic boundary conditions (see Ranocha and Ricchiuto (2024)).
@@ -310,7 +310,7 @@ function create_cache(mesh,
 end
 
 function assemble_system_matrix!(cache, h, D1, D1mat,
-                                 equations::SerreGreenNaghdiEquations1D{BathymetryFlat})
+                                 ::SerreGreenNaghdiEquations1D{BathymetryFlat})
     (; M_h, M_h3_3) = cache
 
     @.. M_h = h
@@ -355,28 +355,6 @@ function assemble_system_matrix!(cache, h, b_x, D1, D1mat,
                      Diagonal(M_h2_bx) * D1mat)
 end
 
-function solve_system_matrix!(dv, system_matrix, rhs,
-                              ::SerreGreenNaghdiEquations1D,
-                              D1, cache)
-    if issparse(system_matrix)
-        (; factorization) = cache
-        cholesky!(factorization, system_matrix; check = false)
-        if issuccess(factorization)
-            scale_by_mass_matrix!(rhs, D1)
-            dv .= factorization \ rhs
-        else
-            # The factorization may fail if the time step is too large
-            # and h becomes negative.
-            fill!(dv, NaN)
-        end
-    else
-        factorization = cholesky!(system_matrix)
-        scale_by_mass_matrix!(rhs, D1)
-        ldiv!(dv, factorization, rhs)
-    end
-    return nothing
-end
-
 # Discretization that conserves
 # - the total water mass (integral of h) as a linear invariant
 # - the total momentum (integral of h v) as a nonlinear invariant for flat bathymetry
@@ -417,7 +395,7 @@ function rhs_sgn_central!(dq, q, equations, source_terms, cache, ::BathymetryFla
     @trixi_timeit timer() "hyperbolic terms" begin
         # Compute all derivatives required below
         (; h, b, h_x, v_x, h2_x, hv_x, v2_x, h2_v_vx_x,
-        h_vx_x, p_x, tmp, M_h, M_h3_3) = cache
+        h_vx_x, p_x, tmp) = cache
 
         @.. b = equations.eta0 - D
         @.. h = eta - b
@@ -499,8 +477,7 @@ function rhs_sgn_upwind!(dq, q, equations, source_terms, cache, ::BathymetryFlat
     @trixi_timeit timer() "hyperbolic terms" begin
         # Compute all derivatives required below
         (; h, b, h_x, v_x, v_x_upwind, h2_x, hv_x, v2_x,
-        h2_v_vx_x, h_vx_x, p_x, tmp,
-        M_h, M_h3_3) = cache
+        h2_v_vx_x, h_vx_x, p_x, tmp) = cache
 
         @.. b = equations.eta0 - D
         @.. h = eta - b
@@ -584,8 +561,7 @@ function rhs_sgn_central!(dq, q, equations, source_terms, cache,
     @trixi_timeit timer() "hyperbolic terms" begin
         # Compute all derivatives required below
         (; h, h_x, v_x, h_hpb_x, b, b_x, hv_x, v2_x,
-        h2_v_vx_x, h_vx_x, p_h, p_x, tmp,
-        M_h_p_h_bx2, M_h3_3, M_h2_bx) = cache
+        h2_v_vx_x, h_vx_x, p_h, p_x, tmp) = cache
         if equations.bathymetry_type isa BathymetryVariable
             (; psi) = cache
         end
@@ -692,8 +668,7 @@ function rhs_sgn_upwind!(dq, q, equations, source_terms, cache,
     @trixi_timeit timer() "hyperbolic terms" begin
         # Compute all derivatives required below
         (; h, h_x, v_x, v_x_upwind, h_hpb_x, b, b_x, hv_x, v2_x,
-        h2_v_vx_x, h_vx_x, p_h, p_0, p_x, tmp,
-        M_h_p_h_bx2, M_h3_3, M_h2_bx) = cache
+        h2_v_vx_x, h_vx_x, p_h, p_0, p_x, tmp) = cache
         if equations.bathymetry_type isa BathymetryVariable
             (; psi) = cache
         end
@@ -806,11 +781,11 @@ end
     return SVector(eta, v, D)
 end
 
-@inline function waterheight_total(q, equations::AbstractSerreGreenNaghdiEquations1D)
+@inline function waterheight_total(q, ::AbstractSerreGreenNaghdiEquations1D)
     return q[1]
 end
 
-@inline function velocity(q, equations::AbstractSerreGreenNaghdiEquations1D)
+@inline function velocity(q, ::AbstractSerreGreenNaghdiEquations1D)
     return q[2]
 end
 
@@ -819,11 +794,11 @@ end
     return equations.eta0 - D
 end
 
-# The entropy/energy takes the whole `q` for every point in space
+# The modified entropy/energy takes the whole `q` for every point in space
 """
-    energy_total_modified(q_global, equations::SerreGreenNaghdiEquations1D, cache)
+    DispersiveShallowWater.energy_total_modified!(e, q_global, equations::SerreGreenNaghdiEquations1D, cache)
 
-Return the modified total energy of the primitive variables `q_global` for the
+Return the modified total energy `e` of the primitive variables `q_global` for the
 [`SerreGreenNaghdiEquations1D`](@ref).
 It contains an additional term containing a
 derivative compared to the usual [`energy_total`](@ref) modeling
@@ -845,24 +820,21 @@ For a [`bathymetry_variable`](@ref) the total modified energy has the additional
 
 `q_global` is a vector of the primitive variables at ALL nodes.
 `cache` needs to hold the SBP operators used by the `solver`.
+
+See also [`energy_total_modified`](@ref).
 """
-function energy_total_modified(q_global,
-                               equations::SerreGreenNaghdiEquations1D,
-                               cache)
+function energy_total_modified!(e, q_global,
+                                equations::SerreGreenNaghdiEquations1D,
+                                cache)
     # unpack physical parameters and SBP operator `D1`
     g = gravity_constant(equations)
     (; D1, h, b, v_x) = cache
 
     # `q_global` is an `ArrayPartition`. It collects the individual arrays for
     # the total water height `eta = h + b` and the velocity `v`.
-    eta, v = q_global.x
-    let D = q_global.x[3]
-        @.. b = equations.eta0 - D
-        @.. h = eta - b
-    end
-
-    N = length(v)
-    e = zeros(eltype(q_global), N)
+    eta, v, D = q_global.x
+    @.. b = equations.eta0 - D
+    @.. h = eta - b
 
     # 1/2 g eta^2 + 1/2 h v^2 + 1/6 h^3 w^2
     # and + 1/8 h (v b_x)^2 for full bathymetry without mild-slope approximation