Added temptative example elixir for coupled GlmMhd equations.

This requires the modified callback for the GLM cleaning speed.

examples/structured_2d_dgsem/elixir_mhd_coupled.jl

@@ -0,0 +1,138 @@
+using OrdinaryDiffEq
+using Trixi
+
+###############################################################################
+# Coupled semidiscretization of two ideal glmMhd systems using converter functions such that
+# they are also coupled across the domain boundaries to generate a periodic system.
+#
+# In this elixir, we have a square domain that is divided into a left and right half.
+# On each half of the domain, a completely independent SemidiscretizationHyperbolic is created for the
+# linear ideal glmMhd equations. The four systems are coupled in the x and y-direction.
+# For a high-level overview, see also the figure below:
+#
+# (-2,  2)                                   ( 2,  2)
+#     ┌────────────────────┬────────────────────┐
+#     │    ↑ periodic ↑    │    ↑ periodic ↑    │
+#     │                    │                    │
+#     │     =========      │     =========      │
+#     │     system #1      │     system #2      │
+#     │     =========      │     =========      │
+#     │                    │                    │
+#     │<-- coupled         │<-- coupled         │
+#     │         coupled -->│         coupled -->│
+#     │                    │                    │
+#     │    ↓ periodic ↓    │    ↓ periodic ↓    │
+#     └────────────────────┴────────────────────┘
+# (-2, -2)                                   ( 2, -2)
+
+
+equations = IdealGlmMhdEquations2D(1.4)
+
+cells_per_dimension = (32, 64)
+
+volume_flux = (flux_hindenlang_gassner, flux_nonconservative_powell)
+solver = DGSEM(polydeg = 3,
+               surface_flux = (flux_lax_friedrichs, flux_nonconservative_powell),
+               volume_integral = VolumeIntegralFluxDifferencing(volume_flux))
+
+function initial_condition_constant(x, t, equations::IdealGlmMhdEquations2D)
+  rho = 1.0
+  v1 = 0.0
+  v2 = 0.0
+  v3 = 0.0
+  p = rho^equations.gamma
+  B1 = 0.0
+  B2 = 0.0
+  B3 = 0.0
+  psi = 0.0
+  return prim2cons(SVector(rho, v1, v2, v3, p, B1, B2, B3, psi), equations)
+end
+
+###########
+# system #1
+###########
+
+initial_condition1 = initial_condition_constant
+coordinates_min1 = (-2.0, -2.0)
+coordinates_max1 = ( 0.0,  2.0)
+mesh1 = StructuredMesh(cells_per_dimension,
+                       coordinates_min1,
+                       coordinates_max1)
+
+coupling_function1 = (x, u, equations_other, equations_own) -> u
+boundary_conditions1 = (
+                       x_neg=BoundaryConditionCoupled(2, (:end, :i_forward), Float64, coupling_function1),
+                       x_pos=BoundaryConditionCoupled(2, (:begin, :i_forward), Float64, coupling_function1),
+                       y_neg=boundary_condition_periodic,
+                       y_pos=boundary_condition_periodic,
+                      )
+
+semi1 = SemidiscretizationHyperbolic(mesh1, equations, initial_condition1, solver,
+                                     boundary_conditions=boundary_conditions1)
+
+###########
+# system #2
+###########
+
+initial_condition2 = initial_condition_constant
+coordinates_min2 = ( 0.0, -2.0)
+coordinates_max2 = ( 2.0,  2.0)
+mesh2 = StructuredMesh(cells_per_dimension,
+                       coordinates_min2,
+                       coordinates_max2)
+
+coupling_function2 = (x, u, equations_other, equations_own) -> u
+boundary_conditions2 = (
+                       x_neg=BoundaryConditionCoupled(1, (:end, :i_forward), Float64, coupling_function2),
+                       x_pos=BoundaryConditionCoupled(1, (:begin, :i_forward), Float64, coupling_function2),
+                       y_neg=boundary_condition_periodic,
+                       y_pos=boundary_condition_periodic,
+                      )
+
+semi2 = SemidiscretizationHyperbolic(mesh2, equations, initial_condition2, solver,
+                                     boundary_conditions=boundary_conditions2)
+
+# Create a semidiscretization that bundles all the semidiscretizations.
+semi = SemidiscretizationCoupled(semi1, semi2)
+
+###############################################################################
+# ODE solvers, callbacks etc.
+
+tspan = (0.0, 0.1)
+ode = semidiscretize(semi, tspan)
+
+summary_callback = SummaryCallback()
+
+analysis_interval = 100
+
+analysis_callback1 = AnalysisCallback(semi1, interval=100)
+analysis_callback2 = AnalysisCallback(semi2, interval=100)
+analysis_callback = AnalysisCallbackCoupled(semi, analysis_callback1, analysis_callback2)
+
+alive_callback = AliveCallback(analysis_interval=analysis_interval)
+
+save_solution = SaveSolutionCallback(interval=50,
+                                     save_initial_solution=true,
+                                     save_final_solution=true,
+                                     solution_variables=cons2prim)
+
+cfl = 1.0
+
+stepsize_callback = StepsizeCallback(cfl=cfl)
+
+glm_speed_callback = GlmSpeedCallback(glm_scale=0.5, cfl=cfl, semi_indices=tuple(1))
+
+callbacks = CallbackSet(summary_callback,
+                        analysis_callback, alive_callback,
+                        save_solution,
+                        stepsize_callback,
+                        glm_speed_callback)
+
+
+###############################################################################
+# run the simulation
+
+sol = solve(ode, CarpenterKennedy2N54(williamson_condition=false),
+            dt=0.01, # solve needs some value here but it will be overwritten by the stepsize_callback
+            save_everystep=false, callback=callbacks);
+summary_callback() # print the timer summary