diff --git a/examples/tree_2d_dgsem/elixir_mhdmultiion_ec.jl b/examples/tree_2d_dgsem/elixir_mhdmultiion_ec.jl index d9b66496b1..907ac5d963 100644 --- a/examples/tree_2d_dgsem/elixir_mhdmultiion_ec.jl +++ b/examples/tree_2d_dgsem/elixir_mhdmultiion_ec.jl @@ -9,8 +9,15 @@ equations = IdealGlmMhdMultiIonEquations2D(gammas = (1.4, 1.667), initial_condition = initial_condition_weak_blast_wave +# Entropy conservative numerical fluxes volume_flux = (flux_ruedaramirez_etal, flux_nonconservative_ruedaramirez_etal) surface_flux = (flux_ruedaramirez_etal, flux_nonconservative_ruedaramirez_etal) +# For provably entropy-stable surface fluxes, use +# surface_flux = (FluxPlusDissipation(flux_ruedaramirez_etal, DissipationEntropyStable()), +# flux_nonconservative_ruedaramirez_etal) +# For a standard local lax-friedrichs surface flux, use +# surface_flux = (flux_lax_friedrichs, flux_nonconservative_central) + solver = DGSEM(polydeg = 3, surface_flux = surface_flux, volume_integral = VolumeIntegralFluxDifferencing(volume_flux)) diff --git a/src/Trixi.jl b/src/Trixi.jl index cb50ca2e7d..e47d60d5dc 100644 --- a/src/Trixi.jl +++ b/src/Trixi.jl @@ -188,6 +188,7 @@ export flux, flux_central, flux_lax_friedrichs, flux_hll, flux_hllc, flux_hlle, flux_chan_etal, flux_nonconservative_chan_etal, flux_winters_etal, hydrostatic_reconstruction_audusse_etal, flux_nonconservative_audusse_etal, FluxPlusDissipation, DissipationGlobalLaxFriedrichs, DissipationLocalLaxFriedrichs, + DissipationEntropyStable, FluxLaxFriedrichs, max_abs_speed_naive, FluxHLL, min_max_speed_naive, min_max_speed_davis, min_max_speed_einfeldt, FluxLMARS, diff --git a/src/equations/ideal_glm_mhd_multiion.jl b/src/equations/ideal_glm_mhd_multiion.jl index c76351a375..80db194abb 100644 --- a/src/equations/ideal_glm_mhd_multiion.jl +++ b/src/equations/ideal_glm_mhd_multiion.jl @@ -263,4 +263,171 @@ end return SVector(cons) end + +# Specialization of DissipationEntropyStable for the multi-ion GLM-MHD equations +# See +# - A. Rueda-Ramírez, A. Sikstel, G. Gassner, An Entropy-Stable Discontinuous Galerkin Discretization +# of the Ideal Multi-Ion Magnetohydrodynamics System (2024). Journal of Computational Physics. +# [DOI: 10.1016/j.jcp.2024.113655](https://doi.org/10.1016/j.jcp.2024.113655). +@inline function (dissipation::DissipationEntropyStable)(u_ll, u_rr, + orientation_or_normal_direction, + equations::AbstractIdealGlmMhdMultiIonEquations) + @unpack gammas = equations + λ = dissipation.max_abs_speed(u_ll, u_rr, orientation_or_normal_direction, + equations) + + w_ll = cons2entropy(u_ll, equations) + w_rr = cons2entropy(u_rr, equations) + prim_ll = cons2prim(u_ll, equations) + prim_rr = cons2prim(u_rr, equations) + B1_ll, B2_ll, B3_ll = magnetic_field(u_ll, equations) + B1_rr, B2_rr, B3_rr = magnetic_field(u_rr, equations) + psi_ll = divergence_cleaning_field(u_ll, equations) + psi_rr = divergence_cleaning_field(u_rr, equations) + + # Some global averages + B1_avg = 0.5f0 * (B1_ll + B1_rr) + B2_avg = 0.5f0 * (B2_ll + B2_rr) + B3_avg = 0.5f0 * (B3_ll + B3_rr) + psi_avg = 0.5f0 * (psi_ll + psi_rr) + + dissipation = zero(MVector{nvariables(equations), eltype(u_ll)}) + + beta_plus_ll = 0 + beta_plus_rr = 0 + # Get the lumped dissipation for all components + for k in eachcomponent(equations) + rho_ll, v1_ll, v2_ll, v3_ll, p_ll = get_component(k, prim_ll, equations) + rho_rr, v1_rr, v2_rr, v3_rr, p_rr = get_component(k, prim_rr, equations) + + w1_ll, w2_ll, w3_ll, w4_ll, w5_ll = get_component(k, w_ll, equations) + w1_rr, w2_rr, w3_rr, w4_rr, w5_rr = get_component(k, w_rr, equations) + + # Auxiliary variables + beta_ll = 0.5f0 * rho_ll / p_ll + beta_rr = 0.5f0 * rho_rr / p_rr + vel_norm_ll = v1_ll^2 + v2_ll^2 + v3_ll^2 + vel_norm_rr = v1_rr^2 + v2_rr^2 + v3_rr^2 + + # Mean variables + rho_ln = ln_mean(rho_ll, rho_rr) + beta_ln = ln_mean(beta_ll, beta_rr) + rho_avg = 0.5f0 * (rho_ll + rho_rr) + v1_avg = 0.5f0 * (v1_ll + v1_rr) + v2_avg = 0.5f0 * (v2_ll + v2_rr) + v3_avg = 0.5f0 * (v3_ll + v3_rr) + beta_avg = 0.5f0 * (beta_ll + beta_rr) + tau = 1 / (beta_ll + beta_rr) + p_mean = 0.5f0 * rho_avg / beta_avg + p_star = 0.5f0 * rho_ln / beta_ln + vel_norm_avg = 0.5f0 * (vel_norm_ll + vel_norm_rr) + vel_avg_norm = v1_avg^2 + v2_avg^2 + v3_avg^2 + E_bar = p_star / (gammas[k] - 1) + + 0.5f0 * rho_ln * (2 * vel_avg_norm - vel_norm_avg) + + h11 = rho_ln + h12 = rho_ln * v1_avg + h13 = rho_ln * v2_avg + h14 = rho_ln * v3_avg + h15 = E_bar + d1 = -0.5f0 * λ * + (h11 * (w1_rr - w1_ll) + + h12 * (w2_rr - w2_ll) + + h13 * (w3_rr - w3_ll) + + h14 * (w4_rr - w4_ll) + + h15 * (w5_rr - w5_ll)) + + h21 = h12 + h22 = rho_ln * v1_avg^2 + p_mean + h23 = h21 * v2_avg + h24 = h21 * v3_avg + h25 = (E_bar + p_mean) * v1_avg + d2 = -0.5f0 * λ * + (h21 * (w1_rr - w1_ll) + + h22 * (w2_rr - w2_ll) + + h23 * (w3_rr - w3_ll) + + h24 * (w4_rr - w4_ll) + + h25 * (w5_rr - w5_ll)) + + h31 = h13 + h32 = h23 + h33 = rho_ln * v2_avg^2 + p_mean + h34 = h31 * v3_avg + h35 = (E_bar + p_mean) * v2_avg + d3 = -0.5f0 * λ * + (h31 * (w1_rr - w1_ll) + + h32 * (w2_rr - w2_ll) + + h33 * (w3_rr - w3_ll) + + h34 * (w4_rr - w4_ll) + + h35 * (w5_rr - w5_ll)) + + h41 = h14 + h42 = h24 + h43 = h34 + h44 = rho_ln * v3_avg^2 + p_mean + h45 = (E_bar + p_mean) * v3_avg + d4 = -0.5f0 * λ * + (h41 * (w1_rr - w1_ll) + + h42 * (w2_rr - w2_ll) + + h43 * (w3_rr - w3_ll) + + h44 * (w4_rr - w4_ll) + + h45 * (w5_rr - w5_ll)) + + h51 = h15 + h52 = h25 + h53 = h35 + h54 = h45 + h55 = ((p_star^2 / (gammas[k] - 1) + E_bar * E_bar) / rho_ln + + + vel_avg_norm * p_mean) + d5 = -0.5f0 * λ * + (h51 * (w1_rr - w1_ll) + + h52 * (w2_rr - w2_ll) + + h53 * (w3_rr - w3_ll) + + h54 * (w4_rr - w4_ll) + + h55 * (w5_rr - w5_ll)) + + beta_plus_ll += beta_ll + beta_plus_rr += beta_rr + + set_component!(dissipation, k, d1, d2, d3, d4, d5, equations) + end + + # Set the magnetic field and psi terms + h_B_psi = 1 / (beta_plus_ll + beta_plus_rr) + + # diagonal entries + dissipation[1] = -0.5f0 * λ * h_B_psi * (w_rr[1] - w_ll[1]) + dissipation[2] = -0.5f0 * λ * h_B_psi * (w_rr[2] - w_ll[2]) + dissipation[3] = -0.5f0 * λ * h_B_psi * (w_rr[3] - w_ll[3]) + dissipation[end] = -0.5f0 * λ * h_B_psi * (w_rr[end] - w_ll[end]) + # Off-diagonal entries + for k in eachcomponent(equations) + _, _, _, _, w5_ll = get_component(k, w_ll, equations) + _, _, _, _, w5_rr = get_component(k, w_rr, equations) + + dissipation[1] -= 0.5f0 * λ * h_B_psi * B1_avg * (w5_rr - w5_ll) + dissipation[2] -= 0.5f0 * λ * h_B_psi * B2_avg * (w5_rr - w5_ll) + dissipation[3] -= 0.5f0 * λ * h_B_psi * B3_avg * (w5_rr - w5_ll) + dissipation[end] -= 0.5f0 * λ * h_B_psi * psi_avg * (w5_rr - w5_ll) + + # Dissipation for the energy equation of species k depending on w_1, w_2, w_3 and w_end + ind_E = 3 + (k - 1) * 5 + 5 + dissipation[ind_E] -= 0.5f0 * λ * h_B_psi * B1_avg * (w_rr[1] - w_ll[1]) + dissipation[ind_E] -= 0.5f0 * λ * h_B_psi * B2_avg * (w_rr[2] - w_ll[2]) + dissipation[ind_E] -= 0.5f0 * λ * h_B_psi * B3_avg * (w_rr[3] - w_ll[3]) + dissipation[ind_E] -= 0.5f0 * λ * h_B_psi * psi_avg * (w_rr[end] - w_ll[end]) + + # Dissipation for the energy equation of all ion species depending on w_5 + for kk in eachcomponent(equations) + ind_E = 3 + (kk - 1) * 5 + 5 + dissipation[ind_E] -= 0.5f0 * λ * + (h_B_psi * + (B1_avg^2 + B2_avg^2 + B3_avg^2 + psi_avg^2)) * + (w5_rr - w5_ll) + end + end + + return dissipation +end end diff --git a/src/equations/numerical_fluxes.jl b/src/equations/numerical_fluxes.jl index ea75b99b7f..976bc64acb 100644 --- a/src/equations/numerical_fluxes.jl +++ b/src/equations/numerical_fluxes.jl @@ -221,6 +221,46 @@ See [`FluxLaxFriedrichs`](@ref). """ const flux_lax_friedrichs = FluxLaxFriedrichs() +""" + DissipationEntropyStable(max_abs_speed=max_abs_speed_naive) + +Create a local Lax-Friedrichs-type dissipation operator that is provably entropy stable. This operator +must be used together with an entropy-conservative two-point flux function (e.g., `flux_ec`) to yield +an entropy-stable surface flux. The surface flux function can be initialized as: +``` +flux_es = FluxPlusDissipation(flux_ec, DissipationEntropyStable()) +``` + +In particular, the numerical flux has the form +```math +f^{ES} = f^{EC} + \frac{1}{2} λ_{max} H (w_r - w_l), +```` +where ``f^{EC}`` is the entropy-conservative two-point flux function (computed with, e.g., `flux_ec`), ``λ_{max}`` +is the maximum wave speed estimated as `max_abs_speed(u_l, u_r, orientation_or_normal_direction, equations)`, +defaulting to [`max_abs_speed_naive`](@ref), ``H`` is a symmetric positive-definite dissipation matrix that +depends on the left and right states `u_l` and `u_r`, and ``(w_r - w_l)`` is the jump in entropy variables. +Ideally, ``H (w_r - w_l) = (u_r - u_l)``, such that the dissipation operator is consistent with the local +Lax-Friedrichs dissipation. + +The entropy-stable dissipation operator is computed with the function +`function (dissipation::DissipationEntropyStable)(u_l, u_r, orientation_or_normal_direction, equations)`, +which must be specialized for each equation. + +For the derivation of the dissipation matrix for the multi-ion GLM-MHD equations, see: +- A. Rueda-Ramírez, A. Sikstel, G. Gassner, An Entropy-Stable Discontinuous Galerkin Discretization + of the Ideal Multi-Ion Magnetohydrodynamics System (2024). Journal of Computational Physics. + [DOI: 10.1016/j.jcp.2024.113655](https://doi.org/10.1016/j.jcp.2024.113655). +""" +struct DissipationEntropyStable{MaxAbsSpeed} + max_abs_speed::MaxAbsSpeed +end + +DissipationEntropyStable() = DissipationEntropyStable(max_abs_speed_naive) + +function Base.show(io::IO, d::DissipationEntropyStable) + print(io, "DissipationEntropyStable(", d.max_abs_speed, ")") +end + """ FluxHLL(min_max_speed=min_max_speed_davis) diff --git a/test/test_tree_2d_mhdmultiion.jl b/test/test_tree_2d_mhdmultiion.jl index 2ef8b56e9e..73fc9fb46e 100644 --- a/test/test_tree_2d_mhdmultiion.jl +++ b/test/test_tree_2d_mhdmultiion.jl @@ -55,6 +55,53 @@ EXAMPLES_DIR = joinpath(pathof(Trixi) |> dirname |> dirname, "examples", "tree_2 end end +@trixi_testset "Provably entropy-stable LLF-type fluxes for multi-ion GLM-MHD" begin + @test_trixi_include(joinpath(EXAMPLES_DIR, "elixir_mhdmultiion_ec.jl"), + l2=[ + 0.017668017558288736, + 0.01779783612885502, + 0.027841673842076285, + 0.015603429086471762, + 0.017849042999817964, + 0.01814196379994667, + 0.005478212889809162, + 0.20585517887094282, + 0.021301245733548135, + 0.03018506565829777, + 0.02938517728342881, + 0.01837279433780041, + 0.11810307914710033, + 0.0002962677911603057 + ], + linf=[ + 0.06594754030722516, + 0.06587779693691242, + 0.09451464686853495, + 0.06787230638663028, + 0.08910065803824378, + 0.08828064474448032, + 0.023647579422062297, + 0.8059383650828509, + 0.1224367642558366, + 0.15930418161523857, + 0.15382860284948224, + 0.08695364286964764, + 0.4949375933243716, + 0.003287251595115295 + ], + surface_flux=(FluxPlusDissipation(flux_ruedaramirez_etal, + DissipationEntropyStable()), + flux_nonconservative_ruedaramirez_etal)) + # Ensure that we do not have excessive memory allocations + # (e.g., from type instabilities) + let + t = sol.t[end] + u_ode = sol.u[end] + du_ode = similar(u_ode) + @test (@allocated Trixi.rhs!(du_ode, u_ode, semi, t)) < 1000 + end +end + @trixi_testset "elixir_mhdmultiion_ec.jl with local Lax-Friedrichs at the surface" begin @test_trixi_include(joinpath(EXAMPLES_DIR, "elixir_mhdmultiion_ec.jl"), l2=[ diff --git a/test/test_type.jl b/test/test_type.jl index e26b34cdfc..6b45f3cf60 100644 --- a/test/test_type.jl +++ b/test/test_type.jl @@ -1524,6 +1524,7 @@ isdir(outdir) && rm(outdir, recursive = true) one(RealT), one(RealT), one(RealT)) + dissipation_es = DissipationEntropyStable() orientations = [1, 2] @test eltype(@inferred initial_condition_weak_blast_wave(x, t, equations)) == @@ -1547,6 +1548,8 @@ isdir(outdir) && rm(outdir, recursive = true) @test typeof(@inferred max_abs_speed_naive(u_ll, u_rr, orientation, equations)) == RealT + @test eltype(@inferred dissipation_es(u_ll, u_rr, orientation, equations)) == + RealT end @test eltype(@inferred Trixi.max_abs_speeds(u, equations)) == RealT