HLLC Flux for Ideal MHD 1D

src/equations/ideal_glm_mhd_1d.jl

@@ -259,6 +259,148 @@ Hindenlang and Gassner (2019), extending [`flux_ranocha`](@ref) to the MHD equat
     return SVector(f1, f2, f3, f4, f5, f6, f7, f8)
 end
 
+"""
+    flux_hllc(u_ll, u_rr, orientation, equations::IdealGlmMhdEquations1D)
+
+- Li (2005)
+An HLLC Riemann solver for magneto-hydrodynamics
+[DOI: 10.1016/j.jcp.2004.08.020](https://doi.org/10.1016/j.jcp.2004.08.020).
+"""
+function flux_hllc(u_ll, u_rr, orientation::Integer,
+                   equations::IdealGlmMhdEquations1D)
+    # Unpack left and right states
+    rho_ll, v1_ll, v2_ll, v3_ll, p_ll, B1_ll, B2_ll, B3_ll = cons2prim(u_ll, equations)
+    rho_rr, v1_rr, v2_rr, v3_rr, p_rr, B1_rr, B2_rr, B3_rr = cons2prim(u_rr, equations)
+
+    # Conserved variables
+    rho_v1_ll = u_ll[2]
+    rho_v2_ll = u_ll[3]
+    rho_v3_ll = u_ll[4]
+
+    rho_v1_rr = u_rr[2]
+    rho_v2_rr = u_rr[3]
+    rho_v3_rr = u_rr[4]
+
+    # Obtain left and right fluxes
+    f_ll = flux(u_ll, orientation, equations)
+    f_rr = flux(u_rr, orientation, equations)
+
+    SsL, SsR = min_max_speed_naive(u_ll, u_rr, orientation, equations)
+    sMu_L = SsL - v1_ll
+    sMu_R = SsR - v1_rr
+    if SsL >= 0
+        f1 = f_ll[1]
+        f2 = f_ll[2]
+        f3 = f_ll[3]
+        f4 = f_ll[4]
+        f5 = f_ll[5]
+        f6 = f_ll[6]
+        f7 = f_ll[7]
+        f8 = f_ll[8]
+    elseif SsR <= 0
+        f1 = f_rr[1]
+        f2 = f_rr[2]
+        f3 = f_rr[3]
+        f4 = f_rr[4]
+        f5 = f_rr[5]
+        f6 = f_rr[6]
+        f7 = f_rr[7]
+        f8 = f_rr[8]
+    else
+        # Compute the "HLLC-speed", eq. (14) from paper mentioned above
+        #=
+        SStar = (rho_rr * v1_rr * sMu_R - rho_ll * v1_ll * sMu_L + p_ll - p_rr - B1_ll^2 + B1_rr^2 ) /
+                (rho_rr * sMu_R - rho_ll * sMu_L)
+        =#
+        # Simplification for 1D: B1 is constant
+        SStar = (rho_rr * v1_rr * sMu_R - rho_ll * v1_ll * sMu_L + p_ll - p_rr) /
+                (rho_rr * sMu_R - rho_ll * sMu_L)
+
+        Sdiff = SsR - SsL
+
+        # Compute HLL values for vStar, BStar
+        # These correspond to eq. (28) and (30) from the referenced paper 
+        # and the classic HLL intermediate state given by (2)
+        rho_HLL = (SsR * rho_rr - SsL * rho_ll - (f_rr[1] - f_ll[1])) / Sdiff
+
+        v1Star = (SsR * rho_v1_rr - SsL * rho_v1_ll - (f_rr[2] - f_ll[2])) /
+                 (Sdiff * rho_HLL)
+        v2Star = (SsR * rho_v2_rr - SsL * rho_v2_ll - (f_rr[3] - f_ll[3])) /
+                 (Sdiff * rho_HLL)
+        v3Star = (SsR * rho_v3_rr - SsL * rho_v3_ll - (f_rr[4] - f_ll[4])) /
+                 (Sdiff * rho_HLL)
+
+        #B1Star = (SsR * B1_rr - SsL * B1_ll - (f_rr[6] - f_ll[6])) / Sdiff
+        # 1D B1 = constant => B1_ll = B1_rr = B1Star
+        B1Star = B1_ll
+
+        B2Star = (SsR * B2_rr - SsL * B2_ll - (f_rr[7] - f_ll[7])) / Sdiff
+        B3Star = (SsR * B3_rr - SsL * B3_ll - (f_rr[8] - f_ll[8])) / Sdiff
+        if SsL <= SStar
+            SdiffStar = SsL - SStar
+
+            densStar = rho_ll * sMu_L / SdiffStar # (19)
+
+            mom_1_Star = densStar * SStar # (20)
+            mom_2_Star = densStar * v2_ll -
+                         (B1Star * B2Star - B1_ll * B2_ll) / SdiffStar # (21)
+            mom_3_Star = densStar * v3_ll -
+                         (B1Star * B3Star - B1_ll * B3_ll) / SdiffStar # (22)
+
+            #pstar = rho_ll * sMu_L * (SStar - v1_ll) + p_ll - B1_ll^2 + B1Star^2 # (17)
+            # 1D B1 = constant => B1_ll = B1_rr = B1Star
+            pstar = rho_ll * sMu_L * (SStar - v1_ll) + p_ll # (17)
+
+            enerStar = u_ll[5] * sMu_L / SdiffStar +
+                       (pstar * SStar - p_ll * v1_ll - (B1Star *
+                         (B1Star * v1Star + B2Star * v2Star + B3Star * v3Star) -
+                         B1_ll * (B1_ll * v1_ll + B2_ll * v2_ll + B3_ll * v3_ll))) /
+                       SdiffStar # (23)
+
+            # Classic HLLC update (32)
+            f1 = f_ll[1] + SsL * (densStar - u_ll[1])
+            f2 = f_ll[2] + SsL * (mom_1_Star - u_ll[2])
+            f3 = f_ll[3] + SsL * (mom_2_Star - u_ll[3])
+            f4 = f_ll[4] + SsL * (mom_3_Star - u_ll[4])
+            f5 = f_ll[5] + SsL * (enerStar - u_ll[5])
+            f6 = f_ll[6] + SsL * (B1Star - u_ll[6])
+            f7 = f_ll[7] + SsL * (B2Star - u_ll[7])
+            f8 = f_ll[8] + SsL * (B3Star - u_ll[8])
+        else # SStar <= Ssr
+            SdiffStar = SsR - SStar
+
+            densStar = rho_rr * sMu_R / SdiffStar # (19)
+
+            mom_1_Star = densStar * SStar # (20)
+            mom_2_Star = densStar * v2_rr -
+                         (B1Star * B2Star - B1_rr * B2_rr) / SdiffStar # (21)
+            mom_3_Star = densStar * v3_rr -
+                         (B1Star * B3Star - B1_rr * B3_rr) / SdiffStar # (22)
+
+            #pstar = rho_rr * sMu_R * (SStar - v1_rr) + p_rr - B1_rr^2 + B1Star^2 # (17)
+            # 1D B1 = constant => B1_ll = B1_rr = B1Star
+            pstar = rho_rr * sMu_R * (SStar - v1_rr) + p_rr # (17)
+
+            enerStar = u_rr[5] * sMu_R / SdiffStar +
+                       (pstar * SStar - p_rr * v1_rr - (B1Star *
+                         (B1Star * v1Star + B2Star * v2Star + B3Star * v3Star) -
+                         B1_rr * (B1_rr * v1_rr + B2_rr * v2_rr + B3_rr * v3_rr))) /
+                       SdiffStar # (23)
+
+            # Classic HLLC update (32)           
+            f1 = f_rr[1] + SsR * (densStar - u_rr[1])
+            f2 = f_rr[2] + SsR * (mom_1_Star - u_rr[2])
+            f3 = f_rr[3] + SsR * (mom_2_Star - u_rr[3])
+            f4 = f_rr[4] + SsR * (mom_3_Star - u_rr[4])
+            f5 = f_rr[5] + SsR * (enerStar - u_rr[5])
+            f6 = f_rr[6] + SsR * (B1Star - u_rr[6])
+            f7 = f_rr[7] + SsR * (B2Star - u_rr[7])
+            f8 = f_rr[8] + SsR * (B3Star - u_rr[8])
+        end
+    end
+    return SVector(f1, f2, f3, f4, f5, f6, f7, f8)
+end
+
 # Calculate maximum wave speed for local Lax-Friedrichs-type dissipation
 @inline function max_abs_speed_naive(u_ll, u_rr, orientation::Integer,
                                      equations::IdealGlmMhdEquations1D)

test/test_tree_1d_mhd.jl

@@ -206,6 +206,30 @@ end
     end
 end
 
+@trixi_testset "elixir_mhd_torrilhon_shock_tube.jl (HLLC)" begin
+    @test_trixi_include(joinpath(EXAMPLES_DIR, "elixir_mhd_torrilhon_shock_tube.jl"),
+                        surface_flux=flux_hllc,
+                        l2=[
+                            0.4574266553239646, 0.4794143154876439, 0.3407079689595056,
+                            0.44797768430829343, 0.9206916204424165,
+                            1.3216517820475193e-16, 0.2889748702415378,
+                            0.25529778018020927,
+                        ],
+                        linf=[
+                            1.217943947570543, 0.8868438459815245, 0.878215340656725,
+                            0.9710882819266371, 1.6742759645320984,
+                            2.220446049250313e-16, 0.704710220504591, 0.6562122176458641,
+                        ])
+    # 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_mhd_ryujones_shock_tube.jl" begin
     @test_trixi_include(joinpath(EXAMPLES_DIR, "elixir_mhd_ryujones_shock_tube.jl"),
                         l2=[

test/test_unit.jl

@@ -958,6 +958,7 @@ end
 
     for u in u_values
         @test flux_hlle(u, u, 1, equations) ≈ flux(u, 1, equations)
+        @test flux_hllc(u, u, 1, equations) ≈ flux(u, 1, equations)
     end
 
     equations = IdealGlmMhdEquations2D(1.4, 5.0) #= c_h =#