Schulz-Rinne (2D) Riemann Problem

examples/tree_2d_dgsem/elixir_euler_riemannproblem_quadrants_amr.jl

@@ -0,0 +1,167 @@
+using OrdinaryDiffEqSSPRK
+using Trixi
+
+###############################################################################
+## Semidiscretization of the compressible Euler equations
+
+equations = CompressibleEulerEquations2D(1.4)
+
+# Variant of the 4-quadrant Riemann problem considered in 
+# - Carsten W. Schulz-Rinne:
+#   Classification of the Riemann Problem for Two-Dimensional Gas Dynamics
+#   https://doi.org/10.1137/0524006
+# and 
+# - Carsten W. Schulz-Rinne, James P. Collins, and Harland M. Glaz
+#   Numerical Solution of the Riemann Problem for Two-Dimensional Gas Dynamics
+#   https://doi.org/10.1137/0914082
+function initial_condition_rp(x_, t, equations::CompressibleEulerEquations2D)
+    x, y = x_[1], x_[2]
+
+    if x >= 0.5 && y >= 0.5 # 1st quadrant
+        rho, v1, v2, p = (0.5313, 0.0, 0.0, 0.4)
+    elseif x < 0.5 && y >= 0.5 # 2nd quadrant
+        rho, v1, v2, p = (1.0, 0.7276, 0.0, 1.0)
+    elseif x < 0.5 && y < 0.5 # 3rd quadrant
+        rho, v1, v2, p = (0.8, 0.0, 0.0, 1.0)
+    elseif x >= 0.5 && y < 0.5 # 4th quadrant
+        rho, v1, v2, p = (1.0, 0.0, 0.7276, 1.0)
+    end
+
+    prim = SVector(rho, v1, v2, p)
+    return prim2cons(prim, equations)
+end
+initial_condition = initial_condition_rp
+
+# See Section 2.3 of the reference below for a discussion of robust
+# subsonic inflow/outflow boundary conditions.
+#
+# - Jan-Reneé Carlson (2011)
+#   Inflow/Outflow Boundary Conditions with Application to FUN3D.
+#   [NASA TM 20110022658](https://ntrs.nasa.gov/citations/20110022658)
+@inline function boundary_condition_subsonic(u_inner, orientation::Integer,
+                                             direction, x, t,
+                                             surface_flux_function,
+                                             equations::CompressibleEulerEquations2D)
+    rho_loc, v1_loc, v2_loc, p_loc = cons2prim(u_inner, equations)
+
+    # For subsonic boundary: Take pressure from initial condition
+    p_loc = pressure(initial_condition_rp(x, t, equations), equations)
+
+    prim = SVector(rho_loc, v1_loc, v2_loc, p_loc)
+    u_surface = prim2cons(prim, equations)
+
+    return Trixi.flux(u_surface, orientation, equations)
+end
+
+# The flow is subsonic at all boundaries.
+# For small enough simulation times, the solution remains at the initial condition 
+# *along the boundaries* of quadrants 2, 3, and 4.
+# In quadrants 2 and 4 there are non-zero velocity components (v1 in quadrant 2, v2 in quadrant 4)
+# normal to the boundary, which is troublesome for the `boundary_condition_do_nothing`.
+# Thus, the `boundary_condition_subsonic` are used instead.
+boundary_conditions = (x_neg = boundary_condition_subsonic,
+                       x_pos = boundary_condition_do_nothing,
+                       y_neg = boundary_condition_subsonic,
+                       y_pos = boundary_condition_do_nothing)
+
+coordinates_min = (0.0, 0.0)
+coordinates_max = (1.0, 1.0)
+
+mesh = TreeMesh(coordinates_min, coordinates_max,
+                initial_refinement_level = 4,
+                n_cells_max = 100_000,
+                periodicity = false)
+
+# HLLC flux is strictly required for this problem
+surface_flux = flux_hllc
+volume_flux = flux_ranocha
+
+polydeg = 3
+basis = LobattoLegendreBasis(polydeg)
+shock_indicator = IndicatorHennemannGassner(equations, basis,
+                                            alpha_max = 0.5,
+                                            alpha_min = 0.001,
+                                            alpha_smooth = true,
+                                            variable = Trixi.density)
+
+volume_integral = VolumeIntegralShockCapturingHG(shock_indicator;
+                                                 volume_flux_dg = volume_flux,
+                                                 volume_flux_fv = surface_flux)
+
+solver = DGSEM(polydeg = polydeg, surface_flux = surface_flux,
+               volume_integral = volume_integral)
+
+# Specialized function for computing coefficients of `func`,
+# here the discontinuous `initial_condition_rp`.
+#
+# Shift the outer (i.e., ±1 on the reference element) nodes passed to the 
+# `func` inwards by the smallest amount possible, i.e., [-1 + ϵ, +1 - ϵ].
+# This avoids steep gradients in elements if a discontinuity is right at a cell boundary,
+# i.e., if the jump location `x_jump` is at the position of an interface which is shared by 
+# the nodes x_{e-1}^{(i)} = x_{e}^{(1)}.
+#
+# In particular, this results in the typically desired behaviour for 
+# initial conditions of the form
+#           { u_1, if x <= x_jump
+# u(x, t) = {
+#           { u_2, if x > x_jump
+function Trixi.compute_coefficients!(u, func::typeof(initial_condition_rp), t,
+                                     mesh::TreeMesh{2}, equations, dg::DG, cache)
+    Trixi.@threaded for element in eachelement(dg, cache)
+        for j in eachnode(dg), i in eachnode(dg)
+            x_node = Trixi.get_node_coords(cache.elements.node_coordinates, equations, dg,
+                                           i, j, element)
+            if i == 1 # left boundary node
+                x_node = SVector(nextfloat(x_node[1]), x_node[2])
+            elseif i == nnodes(dg) # right boundary node
+                x_node = SVector(prevfloat(x_node[1]), x_node[2])
+            end
+            if j == 1 # bottom boundary node
+                x_node = SVector(x_node[1], nextfloat(x_node[2]))
+            elseif j == nnodes(dg) # top boundary node
+                x_node = SVector(x_node[1], prevfloat(x_node[2]))
+            end
+
+            u_node = func(x_node, t, equations)
+            set_node_vars!(u, u_node, equations, dg, i, j, element)
+        end
+    end
+end
+
+semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver;
+                                    boundary_conditions = boundary_conditions)
+
+###############################################################################
+## ODE solvers, callbacks etc.
+
+tspan = (0.0, 0.25)
+ode = semidiscretize(semi, tspan)
+
+summary_callback = SummaryCallback()
+
+analysis_interval = 1000
+analysis_callback = AnalysisCallback(semi, interval = analysis_interval)
+
+alive_callback = AliveCallback(analysis_interval = analysis_interval)
+
+stepsize_callback = StepsizeCallback(cfl = 2.4)
+
+amr_indicator = IndicatorLöhner(semi, variable = Trixi.density)
+amr_controller = ControllerThreeLevel(semi, amr_indicator,
+                                      base_level = 3,
+                                      med_level = 5, med_threshold = 0.02,
+                                      max_level = 8, max_threshold = 0.04)
+amr_callback = AMRCallback(semi, amr_controller,
+                           interval = 10,
+                           adapt_initial_condition = true)
+
+callbacks = CallbackSet(summary_callback,
+                        analysis_callback, alive_callback,
+                        amr_callback,
+                        stepsize_callback)
+
+###############################################################################
+## Run the simulation
+
+sol = solve(ode, SSPRK54();
+            dt = 1.0, save_everystep = false, callback = callbacks);

test/test_tree_2d_euler.jl

@@ -1104,6 +1104,32 @@ end
         @test (@allocated Trixi.rhs!(du_ode, u_ode, semi, t)) < 1000
     end
 end
+
+@trixi_testset "elixir_euler_riemannproblem_quadrants_amr.jl" begin
+    @test_trixi_include(joinpath(EXAMPLES_DIR,
+                                 "elixir_euler_riemannproblem_quadrants_amr.jl"),
+                        tspan=(0.0, 0.05),
+                        l2=[
+                            0.1368072821004113,
+                            0.14118941733547452,
+                            0.14118941733547452,
+                            0.517531607013678
+                        ],
+                        linf=[
+                            1.2131375710616266,
+                            1.5874929468585157,
+                            1.587492946858516,
+                            6.190605827791707
+                        ])
+    # 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
 end
 
 end # module