Add Bryan/Fritsch warm bubble test case

LibTrixi.jl/examples/libelixir_tree2d_warm_bubble.jl

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+using LibTrixi
+using OrdinaryDiffEq
+using Trixi
+
+# Warm bubble test case from
+# - Wicker, L. J., and Skamarock, W. C. (1998)
+#   A time-splitting scheme for the elastic equations incorporating 
+#   second-order Runge–Kutta time differencing
+#   [DOI: 10.1175/1520-0493(1998)126%3C1992:ATSSFT%3E2.0.CO;2](https://doi.org/10.1175/1520-0493(1998)126%3C1992:ATSSFT%3E2.0.CO;2)
+# See also
+# - Bryan and Fritsch (2002)
+#   A Benchmark Simulation for Moist Nonhydrostatic Numerical Models
+#   [DOI: 10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2](https://doi.org/10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2)
+# - Carpenter, Droegemeier, Woodward, Hane (1990)
+#   Application of the Piecewise Parabolic Method (PPM) to 
+#   Meteorological Modeling
+#   [DOI: 10.1175/1520-0493(1990)118<0586:AOTPPM>2.0.CO;2](https://doi.org/10.1175/1520-0493(1990)118<0586:AOTPPM>2.0.CO;2)
+struct WarmBubbleSetup
+    # Physical constants
+    g::Float64       # gravity of earth
+    c_p::Float64     # heat capacity for constant pressure (dry air)
+    c_v::Float64     # heat capacity for constant volume (dry air)
+    gamma::Float64   # heat capacity ratio (dry air)
+
+    function WarmBubbleSetup(; g = 9.81, c_p = 1004.0, c_v = 717.0, gamma = c_p / c_v)
+        new(g, c_p, c_v, gamma)
+    end
+end
+
+# Initial condition
+function (setup::WarmBubbleSetup)(x, t, equations::CompressibleEulerEquations2D)
+    @unpack g, c_p, c_v = setup
+
+    # center of perturbation
+    center_x = 10000.0
+    center_z = 2000.0
+    # radius of perturbation
+    radius = 2000.0
+    # distance of current x to center of perturbation
+    r = sqrt((x[1] - center_x)^2 + (x[2] - center_z)^2)
+
+    # perturbation in potential temperature
+    potential_temperature_ref = 300.0
+    potential_temperature_perturbation = 0.0
+    if r <= radius
+        potential_temperature_perturbation = 2 * cospi(0.5 * r / radius)^2
+    end
+    potential_temperature = potential_temperature_ref + potential_temperature_perturbation
+
+    # Exner pressure, solves hydrostatic equation for x[2]
+    exner = 1 - g / (c_p * potential_temperature) * x[2]
+
+    # pressure
+    p_0 = 100_000.0  # reference pressure
+    R = c_p - c_v    # gas constant (dry air)
+    p = p_0 * exner^(c_p / R)
+
+    # temperature
+    T = potential_temperature * exner
+
+    # density
+    rho = p / (R * T)
+
+    v1 = 20.0
+    v2 = 0.0
+    E = c_v * T + 0.5 * (v1^2 + v2^2)
+    return SVector(rho, rho * v1, rho * v2, rho * E)
+end
+
+# Source terms
+@inline function (setup::WarmBubbleSetup)(u, x, t, equations::CompressibleEulerEquations2D)
+    @unpack g = setup
+    rho, _, rho_v2, _ = u
+    return SVector(zero(eltype(u)), zero(eltype(u)), -g * rho, -g * rho_v2)
+end
+
+
+# The function to create the simulation state needs to be named `init_simstate`
+function init_simstate()
+
+    ###############################################################################
+    # semidiscretization of the compressible Euler equations
+    warm_bubble_setup = WarmBubbleSetup()
+
+    equations = CompressibleEulerEquations2D(warm_bubble_setup.gamma)
+
+    boundary_conditions = (x_neg = boundary_condition_periodic,
+                        x_pos = boundary_condition_periodic,
+                        y_neg = boundary_condition_slip_wall,
+                        y_pos = boundary_condition_slip_wall)
+
+    polydeg = 3
+    basis = LobattoLegendreBasis(polydeg)
+
+    # This is a good estimate for the speed of sound in this example.
+    # Other values between 300 and 400 should work as well.
+    surface_flux = FluxLMARS(340.0)
+
+    volume_flux = flux_kennedy_gruber
+    volume_integral = VolumeIntegralFluxDifferencing(volume_flux)
+
+    solver = DGSEM(basis, surface_flux, volume_integral)
+
+    coordinates_min = (0.0, 0.0)
+    coordinates_max = (20_000.0, 10_000.0)
+
+    # Same coordinates as in examples/structured_2d_dgsem/elixir_euler_warm_bubble.jl
+    # However TreeMesh will generate a 20_000 x 20_000 square domain instead
+    mesh = TreeMesh(coordinates_min, coordinates_max,
+                    initial_refinement_level = 6,
+                    n_cells_max = 100_000,
+                    periodicity = (true, false))
+
+    semi = SemidiscretizationHyperbolic(mesh, equations, warm_bubble_setup, solver,
+                                        source_terms = warm_bubble_setup,
+                                        boundary_conditions = boundary_conditions)
+
+    ###############################################################################
+    # ODE solvers, callbacks etc.
+
+    tspan = (0.0, 1000.0)  # 1000 seconds final time
+
+    ode = semidiscretize(semi, tspan)
+
+    summary_callback = SummaryCallback()
+
+    analysis_interval = 1000
+
+    analysis_callback = AnalysisCallback(semi, interval = analysis_interval,
+                                        extra_analysis_errors = (:entropy_conservation_error,))
+
+    alive_callback = AliveCallback(analysis_interval = analysis_interval)
+
+    save_solution = SaveSolutionCallback(interval = analysis_interval,
+                                        save_initial_solution = true,
+                                        save_final_solution = true,
+                                        output_directory = "out_bubble",
+                                        solution_variables = cons2prim)
+
+    @inline function Tpot(u, equations::CompressibleEulerEquations2D)
+        rho, _, _, p = cons2prim(u, equations)
+        exner = (p / 100_000)^(1-inv(warm_bubble_setup.gamma))
+        T = p / rho / (warm_bubble_setup.c_p - warm_bubble_setup.c_v)
+        return T / exner
+    end
+    amr_indicator = IndicatorLöhner(semi, variable = Tpot)
+    amr_controller = ControllerThreeLevel(semi, amr_indicator,
+                                        base_level = 4,
+                                        med_level = 6, med_threshold = 0.0002,
+                                        max_level = 8, max_threshold = 0.0005)
+    amr_callback = AMRCallback(semi, amr_controller,
+                            interval = 50,
+                            adapt_initial_condition = true,
+                            adapt_initial_condition_only_refine = true)
+
+    stepsize_callback = StepsizeCallback(cfl = 1.0)
+
+    callbacks = CallbackSet(summary_callback,
+                            analysis_callback,
+                            alive_callback,
+                            save_solution,
+                            amr_callback,
+                            stepsize_callback)
+
+    ###############################################################################
+    # create OrdinaryDiffEq's time integrator
+    integrator = init(ode,
+                      CarpenterKennedy2N54(williamson_condition=false),
+                      dt=1.0, # solve needs some value here but it will be overwritten by the stepsize_callback
+                      save_everystep=false,
+                      callback=callbacks);
+
+    ###############################################################################
+    # Create simulation state
+    simstate = SimulationState(semi, integrator)
+
+    return simstate
+end