Add dry air warm bubble test case

examples/structured_2d_dgsem/elixir_euler_warm_bubble.jl

@@ -0,0 +1,146 @@
+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
+
+###############################################################################
+# 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)
+
+cells_per_dimension = (64, 32)
+mesh = StructuredMesh(cells_per_dimension, coordinates_min, coordinates_max)
+
+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",
+                                     solution_variables = cons2prim)
+
+stepsize_callback = StepsizeCallback(cfl = 1.0)
+
+callbacks = CallbackSet(summary_callback,
+                        analysis_callback,
+                        alive_callback,
+                        save_solution,
+                        stepsize_callback)
+
+###############################################################################
+# run the simulation
+sol = solve(ode, CarpenterKennedy2N54(williamson_condition = false),
+            maxiters = 1.0e7,
+            dt = 1.0, # solve needs some value here but it will be overwritten by the stepsize_callback
+            save_everystep = false, callback = callbacks);
+
+summary_callback()

examples/tree_2d_dgsem/elixir_euler_warm_bubble.jl

@@ -0,0 +1,150 @@
+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
+
+###############################################################################
+# 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 = 10_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",
+                                     solution_variables = cons2prim)
+
+stepsize_callback = StepsizeCallback(cfl = 1.0)
+
+callbacks = CallbackSet(summary_callback,
+                        analysis_callback,
+                        alive_callback,
+                        save_solution,
+                        stepsize_callback)
+
+###############################################################################
+# run the simulation
+sol = solve(ode, CarpenterKennedy2N54(williamson_condition = false),
+            maxiters = 1.0e7,
+            dt = 1.0, # solve needs some value here but it will be overwritten by the stepsize_callback
+            save_everystep = false, callback = callbacks);
+
+summary_callback()

src/equations/compressible_euler_2d.jl

@@ -809,6 +809,98 @@ end
     return SVector(f1m, f2m, f3m, f4m)
 end
 
+"""
+    FluxLMARS(c)(u_ll, u_rr, orientation_or_normal_direction,
+                 equations::CompressibleEulerEquations2D)
+
+Low Mach number approximate Riemann solver (LMARS) for atmospheric flows using
+an estimate `c` of the speed of sound.
+
+References:
+- Xi Chen et al. (2013)
+  A Control-Volume Model of the Compressible Euler Equations with a Vertical
+  Lagrangian Coordinate
+  [DOI: 10.1175/MWR-D-12-00129.1](https://doi.org/10.1175/mwr-d-12-00129.1)
+"""
+struct FluxLMARS{SpeedOfSound}
+    # Estimate for the speed of sound
+    speed_of_sound::SpeedOfSound
+end
+
+@inline function (flux_lmars::FluxLMARS)(u_ll, u_rr, orientation::Integer,
+                                         equations::CompressibleEulerEquations2D)
+    c = flux_lmars.speed_of_sound
+
+    # Unpack left and right state
+    rho_ll, v1_ll, v2_ll, p_ll = cons2prim(u_ll, equations)
+    rho_rr, v1_rr, v2_rr, p_rr = cons2prim(u_rr, equations)
+
+    if orientation == 1
+        v_ll = v1_ll
+        v_rr = v1_rr
+    else # orientation == 2
+        v_ll = v2_ll
+        v_rr = v2_rr
+    end
+
+    rho = 0.5 * (rho_ll + rho_rr)
+    p = 0.5 * (p_ll + p_rr) - 0.5 * c * rho * (v_rr - v_ll)
+    v = 0.5 * (v_ll + v_rr) - 1 / (2 * c * rho) * (p_rr - p_ll)
+
+    # We treat the energy term analogous to the potential temperature term in the paper by
+    # Chen et al., i.e. we use p_ll and p_rr, and not p
+    if v >= 0
+        f1, f2, f3, f4 = v * u_ll
+        f4 = f4 + p_ll * v
+    else
+        f1, f2, f3, f4 = v * u_rr
+        f4 = f4 + p_rr * v
+    end
+
+    if orientation == 1
+        f2 = f2 + p
+    else # orientation == 2
+        f3 = f3 + p
+    end
+
+    return SVector(f1, f2, f3, f4)
+end
+
+@inline function (flux_lmars::FluxLMARS)(u_ll, u_rr, normal_direction::AbstractVector,
+                                         equations::CompressibleEulerEquations2D)
+    c = flux_lmars.speed_of_sound
+
+    # Unpack left and right state
+    rho_ll, v1_ll, v2_ll, p_ll = cons2prim(u_ll, equations)
+    rho_rr, v1_rr, v2_rr, p_rr = cons2prim(u_rr, equations)
+
+    v_ll = v1_ll * normal_direction[1] + v2_ll * normal_direction[2]
+    v_rr = v1_rr * normal_direction[1] + v2_rr * normal_direction[2]
+
+    # Note that this is the same as computing v_ll and v_rr with a normalized normal vector
+    # and then multiplying v by `norm_` again, but this version is slightly faster.
+    norm_ = norm(normal_direction)
+
+    rho = 0.5 * (rho_ll + rho_rr)
+    p = 0.5 * (p_ll + p_rr) - 0.5 * c * rho * (v_rr - v_ll) / norm_
+    v = 0.5 * (v_ll + v_rr) - 1 / (2 * c * rho) * (p_rr - p_ll) * norm_
+
+    # We treat the energy term analogous to the potential temperature term in the paper by
+    # Chen et al., i.e. we use p_ll and p_rr, and not p
+    if v >= 0
+        f1, f2, f3, f4 = u_ll * v
+        f4 = f4 + p_ll * v
+    else
+        f1, f2, f3, f4 = u_rr * v
+        f4 = f4 + p_rr * v
+    end
+
+    return SVector(f1,
+                   f2 + p * normal_direction[1],
+                   f3 + p * normal_direction[2],
+                   f4)
+end
+
 """
     splitting_vanleer_haenel(u, orientation::Integer,
                              equations::CompressibleEulerEquations2D)