WIP: Well-balanced mortars

examples/p4est_2d_dgsem/elixir_shallowwater_perturbation.jl

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+
+using Downloads: download
+using OrdinaryDiffEq
+using Trixi
+using TrixiShallowWater
+
+###############################################################################
+# semidiscretization of the shallow water equations with a continuous
+# bottom topography function
+
+equations = ShallowWaterEquationsWetDry2D(gravity_constant = 9.812, H0 = 2.1)
+
+function initial_condition_perturbation(x, t, equations::ShallowWaterEquationsWetDry2D)
+    # Calculate primitive variables
+    H = equations.H0
+    v1 = 0.0
+    v2 = 0.0
+
+    x1, x2 = x
+    b = (1.75 / exp(0.5 * ((x1 - 1.0)^2 + (x2 + 1.0)^2))
+         +
+         0.8 / exp(0.5 * ((x1 + 1.0)^2 + (x2 - 1.0)^2))
+         -
+         0.5 / exp(3.5 * ((x1 - 0.4)^2 + (x2 - 0.325)^2)))
+
+    # Waterheight perturbation
+    H = H + 0.5 * exp(-40.0 * ((x[1])^2 + (x[2])^2))
+
+    return prim2cons(SVector(H, v1, v2, b), equations)
+end
+
+initial_condition = initial_condition_perturbation
+
+# Wall BCs
+boundary_condition = Dict(:all => boundary_condition_slip_wall)
+
+###############################################################################
+# Get the DG approximation space
+
+volume_flux = (flux_wintermeyer_etal, flux_nonconservative_wintermeyer_etal)
+
+surface_flux = (FluxHydrostaticReconstruction(flux_hll_chen_noelle, hydrostatic_reconstruction_chen_noelle),
+                flux_nonconservative_chen_noelle)
+
+# Create the solver
+solver = DGSEM(polydeg = 3, surface_flux = surface_flux,
+               volume_integral = VolumeIntegralFluxDifferencing(volume_flux))
+
+###############################################################################
+# This setup is for the curved, split form well-balancedness testing
+
+# Unstructured mesh with 24 cells of the square domain [-1, 1]^2
+mesh_file = Trixi.download("https://gist.githubusercontent.com/efaulhaber/63ff2ea224409e55ee8423b3a33e316a/raw/7db58af7446d1479753ae718930741c47a3b79b7/square_unstructured_2.inp",
+                           joinpath(@__DIR__, "square_unstructured_2.inp"))
+
+# Affine type mapping to take the [-1,1]^2 domain from the mesh file
+# and warp it as described in https://arxiv.org/abs/2012.12040
+# Warping with the coefficient 0.2 is even more extreme.
+function mapping_twist(xi, eta)
+    y = eta + 0.175 * cos(1.5 * pi * xi) * cos(0.5 * pi * eta)
+    x = xi + 0.175 * cos(0.5 * pi * xi) * cos(2.0 * pi * y)
+    return SVector(x, y)
+end
+
+mesh = P4estMesh{2}(mesh_file, polydeg = 3,
+                    mapping = mapping_twist,
+                    initial_refinement_level = 0)
+
+# Create the semi discretization object
+semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver,
+                                    boundary_conditions = boundary_condition)
+
+###############################################################################
+# ODE solvers, callbacks, etc.
+
+tspan = (0.0, 0.5)
+ode = semidiscretize(semi, tspan)
+
+###############################################################################
+
+summary_callback = SummaryCallback()
+
+analysis_interval = 100
+analysis_callback = AnalysisCallback(semi, interval = analysis_interval)
+
+alive_callback = AliveCallback(analysis_interval = analysis_interval)
+
+save_solution = SaveSolutionCallback(dt = 0.04,
+                                     save_initial_solution = true,
+                                     save_final_solution = true)
+
+# Define a better variable to use in the AMR indicator
+@inline function total_water_height(u, equations::ShallowWaterEquationsWetDry2D)
+  return u[1] + u[4]
+end
+
+amr_controller = ControllerThreeLevel(semi, IndicatorMax(semi, variable = total_water_height),
+                                      base_level = 0,
+                                      med_level = 1, med_threshold = 2.01,
+                                      max_level = 4, max_threshold = 2.15)
+
+amr_callback = AMRCallback(semi, amr_controller,
+                           interval = 1,
+                           adapt_initial_condition = false, # to setup discontinuous bottom easier
+                           adapt_initial_condition_only_refine = false)
+
+stepsize_callback = StepsizeCallback(cfl = 1.0)
+
+callbacks = CallbackSet(summary_callback,
+                        analysis_callback,
+                        alive_callback,
+                        save_solution,
+                        amr_callback,
+                        stepsize_callback)
+
+###############################################################################
+# run the simulation
+sol = solve(ode, CarpenterKennedy2N54(williamson_condition = false),
+            dt = 1.0, # solve needs some value here but it will be overwritten by the stepsize_callback
+            # adaptive = false,
+            save_everystep = false, callback = callbacks);
+summary_callback() # print the timer summary

examples/p4est_2d_dgsem/elixir_shallowwater_perturbation_wet_dry.jl

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+
+using Downloads: download
+using OrdinaryDiffEq
+using Trixi
+using TrixiShallowWater
+
+# TODO: This elixir is for debugging purposes of the AMR + well-balanced mortars.
+# Currently this crashes very early in the simulation. ONe possible explanation is that
+# the projection / interpolation of the solution as it is refined / coarsened may need
+# to use a similar strategy as the mortar projection where we ensure that a constant solution
+# is projected on the dry elements. Further investigation is needed to see if this is indeed
+# the case.
+
+###############################################################################
+# semidiscretization of the shallow water equations with a continuous
+# bottom topography function
+
+equations = ShallowWaterEquationsWetDry2D(gravity_constant = 9.812, H0 = 1.235)
+
+function initial_condition_perturbation(x, t, equations::ShallowWaterEquationsWetDry2D)
+    # Calculate primitive variables
+    H = equations.H0
+    v1 = 0.0
+    v2 = 0.0
+
+    x1, x2 = x
+    b = (1.5 / exp(0.5 * ((x1 - 1.0)^2 + (x2 + 1.0)^2))
+         +
+         0.8 / exp(0.5 * ((x1 + 1.0)^2 + (x2 - 1.0)^2))
+         -
+         0.5 / exp(3.5 * ((x1 - 0.4)^2 + (x2 - 0.325)^2)))
+
+    # for testing
+    # b = zero(eltype(x))
+
+    H = max(equations.H0, b + equations.threshold_limiter)
+
+    return prim2cons(SVector(H, v1, v2, b), equations)
+end
+
+initial_condition = initial_condition_perturbation
+
+# Wall BCs
+boundary_condition = Dict(:all => boundary_condition_slip_wall)
+
+###############################################################################
+# Get the DG approximation space
+
+volume_flux = (flux_wintermeyer_etal, flux_nonconservative_wintermeyer_etal)
+
+surface_flux = (FluxHydrostaticReconstruction(flux_hll_chen_noelle, hydrostatic_reconstruction_chen_noelle),
+                flux_nonconservative_chen_noelle)
+
+# # Create the solver (for fully wet!)
+# solver = DGSEM(polydeg = 3, surface_flux = surface_flux,
+#                volume_integral = VolumeIntegralFluxDifferencing(volume_flux))
+
+# Create the solver
+basis = LobattoLegendreBasis(3)
+
+indicator_sc = IndicatorHennemannGassnerShallowWater(equations, basis,
+                                                     alpha_max = 0.5,
+                                                     alpha_min = 0.001,
+                                                     alpha_smooth = true,
+                                                     variable = Trixi.waterheight) # waterheight_pressure)
+volume_integral = VolumeIntegralShockCapturingHG(indicator_sc;
+                                                 volume_flux_dg = volume_flux,
+                                                 volume_flux_fv = surface_flux)
+
+solver = DGSEM(basis, surface_flux, volume_integral)
+
+###############################################################################
+# This setup is for the curved, split form well-balancedness testing
+
+# Unstructured mesh with 24 cells of the square domain [-1, 1]^2
+mesh_file = Trixi.download("https://gist.githubusercontent.com/efaulhaber/63ff2ea224409e55ee8423b3a33e316a/raw/7db58af7446d1479753ae718930741c47a3b79b7/square_unstructured_2.inp",
+                           joinpath(@__DIR__, "square_unstructured_2.inp"))
+
+# Affine type mapping to take the [-1,1]^2 domain from the mesh file
+# and warp it as described in https://arxiv.org/abs/2012.12040
+# Warping with the coefficient 0.2 is even more extreme.
+function mapping_twist(xi, eta)
+    y = eta + 0.15 * cos(1.5 * pi * xi) * cos(0.5 * pi * eta)
+    x = xi + 0.15 * cos(0.5 * pi * xi) * cos(2.0 * pi * y)
+    return SVector(x, y)
+end
+
+mesh = P4estMesh{2}(mesh_file, polydeg = 3,
+                    mapping = mapping_twist,
+                    initial_refinement_level = 0)
+
+# Refine bottom left quadrant of each tree to level 2
+function refine_fn(p4est, which_tree, quadrant)
+  quadrant_obj = unsafe_load(quadrant)
+  if quadrant_obj.x == 0 && quadrant_obj.y == 0 && quadrant_obj.level < 3
+      # return true (refine)
+      return Cint(1)
+  else
+      # return false (don't refine)
+      return Cint(0)
+  end
+end
+
+# Refine recursively until each bottom left quadrant of a tree has level 3
+# The mesh will be rebalanced before the simulation starts
+refine_fn_c = @cfunction(refine_fn, Cint,
+                       (Ptr{Trixi.p4est_t}, Ptr{Trixi.p4est_topidx_t},
+                        Ptr{Trixi.p4est_quadrant_t}))
+Trixi.refine_p4est!(mesh.p4est, true, refine_fn_c, C_NULL)
+
+# Create the semi discretization object
+semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver,
+                                    boundary_conditions = boundary_condition)
+
+###############################################################################
+# ODE solvers, callbacks, etc.
+
+tspan = (0.0, 0.5)
+ode = semidiscretize(semi, tspan)
+
+function initial_condition_discontinuous_well_balancedness(x, t, element_id, equations::ShallowWaterEquationsWetDry2D)
+    # Set the background values for velocity
+    H = equations.H0
+    v1 = zero(H)
+    v2 = zero(H)
+
+    x1, x2 = x
+    b = (1.75 / exp(0.6 * ((x1 - 1.0)^2 + (x2 + 1.0)^2))
+        +
+        0.8 / exp(0.5 * ((x1 + 1.0)^2 + (x2 - 1.0)^2))
+        -
+        0.5 / exp(3.5 * ((x1 - 0.4)^2 + (x2 - 0.325)^2)))
+
+    # Setup a discontinuous bottom topography using the element id number
+    if element_id > 207 && element_id < 300 # for the forced hanging node grid with with < 3 in function above
+        b = (0.75 / exp(0.5 * ((x1 - 1.0)^2 + (x2 + 1.0)^2))
+            +
+            0.4 / exp(0.5 * ((x1 + 1.0)^2 + (x2 - 1.0)^2))
+            -
+            0.25 / exp(3.5 * ((x1 - 0.4)^2 + (x2 - 0.325)^2)))
+    end
+
+    # Put in a discontinous purturbation using the element number
+    if element_id in [232, 224, 225, 226, 227, 228, 229, 230]
+        H = H + 0.3
+    end
+
+    # Clip the initialization to avoid negative water heights and division by zero
+    h = max(equations.threshold_limiter, H - b)
+
+    # Return the conservative variables
+    return SVector(h, h * v1, h * v2, b)
+end
+
+# point to the data we want to augment
+u = Trixi.wrap_array(ode.u0, semi)
+# reset the initial condition
+for element in eachelement(semi.solver, semi.cache)
+  for j in eachnode(semi.solver), i in eachnode(semi.solver)
+      x_node = Trixi.get_node_coords(semi.cache.elements.node_coordinates, equations, semi.solver, i, j, element)
+      u_node = initial_condition_discontinuous_well_balancedness(x_node, first(tspan), element, equations)
+      Trixi.set_node_vars!(u, u_node, equations, semi.solver, i, j, element)
+  end
+end
+
+###############################################################################
+
+summary_callback = SummaryCallback()
+
+analysis_interval = 1000
+analysis_callback = AnalysisCallback(semi, interval = analysis_interval)
+
+alive_callback = AliveCallback(analysis_interval = analysis_interval)
+
+save_solution = SaveSolutionCallback(interval=20, #dt = 0.04,
+                                     save_initial_solution = true,
+                                     save_final_solution = true)
+
+# # Define a better variable to use in the AMR indicator
+# @inline function total_water_height(u, equations::ShallowWaterEquationsWetDry2D)
+#   return u[1] + u[4]
+# end
+
+# amr_controller = ControllerThreeLevel(semi, IndicatorMax(semi, variable = total_water_height),
+#                                       base_level = 1,
+#                                       med_level = 2, med_threshold = 1.225, # 2.01,
+#                                       max_level = 4, max_threshold = 1.245) # 2.15)
+
+# amr_callback = AMRCallback(semi, amr_controller,
+#                            interval = 1,
+#                            adapt_initial_condition = false,
+#                            adapt_initial_condition_only_refine = false)
+
+stepsize_callback = StepsizeCallback(cfl = 0.2)
+
+callbacks = CallbackSet(summary_callback,
+                        analysis_callback,
+                        alive_callback,
+                        save_solution,
+                        # amr_callback,
+                        stepsize_callback)
+
+stage_limiter! = PositivityPreservingLimiterShallowWater(variables = (Trixi.waterheight,))
+
+###############################################################################
+# run the simulation
+sol = solve(ode, SSPRK43(stage_limiter!),
+            dt = 1.0, # solve needs some value here but it will be overwritten by the stepsize_callback
+            adaptive = false,
+            save_everystep = false, callback = callbacks);
+summary_callback() # print the timer summary