Add 3D Cartesian Shallow Water Equations (#36)

Co-authored-by: Tristan Montoya <montoya.tristan@gmail.com>

Project.toml

@@ -6,6 +6,7 @@ version = "0.1.0-DEV"
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 MuladdMacro = "46d2c3a1-f734-5fdb-9937-b9b9aeba4221"
+LoopVectorization = "bdcacae8-1622-11e9-2a5c-532679323890"
 NLsolve = "2774e3e8-f4cf-5e23-947b-6d7e65073b56"
 Reexport = "189a3867-3050-52da-a836-e630ba90ab69"
 Static = "aedffcd0-7271-4cad-89d0-dc628f76c6d3"
@@ -17,6 +18,7 @@ Trixi = "a7f1ee26-1774-49b1-8366-f1abc58fbfcb"
 [compat]
 LinearAlgebra = "1"
 MuladdMacro = "0.2.2"
+LoopVectorization = "0.12.118"
 NLsolve = "4.5.1"
 Reexport = "1.0"
 Static = "0.8, 1"

examples/elixir_euler_spherical_advection_cartesian.jl

@@ -57,7 +57,8 @@ end
 
 # Source term function to transform the Euler equations into the linear advection equations with variable advection velocity
 function source_terms_convert_to_linear_advection(u, du, x, t,
-                                                  equations::CompressibleEulerEquations3D)
+                                                  equations::CompressibleEulerEquations3D,
+                                                  normal_direction)
     v1 = u[2] / u[1]
     v2 = u[3] / u[1]
     v3 = u[4] / u[1]
@@ -70,45 +71,17 @@ function source_terms_convert_to_linear_advection(u, du, x, t,
     return SVector(0.0, s2, s3, s4, s5)
 end
 
-# Custom RHS that applies a source term that depends on du (used to convert the 3D Euler equations into the 3D linear advection
-# equations with position-dependent advection speed)
-function rhs_semi_custom!(du_ode, u_ode, semi, t)
-    # Compute standard Trixi RHS
-    Trixi.rhs!(du_ode, u_ode, semi, t)
-
-    # Now apply the custom source term
-    Trixi.@trixi_timeit Trixi.timer() "custom source term" begin
-        @unpack solver, equations, cache = semi
-        @unpack node_coordinates = cache.elements
-
-        # Wrap the solution and RHS
-        u = Trixi.wrap_array(u_ode, semi)
-        du = Trixi.wrap_array(du_ode, semi)
-
-        Trixi.@threaded for element in eachelement(solver, cache)
-            for j in eachnode(solver), i in eachnode(solver)
-                u_local = Trixi.get_node_vars(u, equations, solver, i, j, element)
-                du_local = Trixi.get_node_vars(du, equations, solver, i, j, element)
-                x_local = Trixi.get_node_coords(node_coordinates, equations, solver,
-                                                i, j, element)
-                source = source_terms_convert_to_linear_advection(u_local, du_local,
-                                                                  x_local, t, equations)
-                Trixi.add_to_node_vars!(du, source, equations, solver, i, j, element)
-            end
-        end
-    end
-end
-
 initial_condition = initial_condition_advection_sphere
 
 element_local_mapping = false
 
-mesh = TrixiAtmo.P4estMeshCubedSphere2D(5, 1.0, polydeg = polydeg,
-                                        initial_refinement_level = 0,
-                                        element_local_mapping = element_local_mapping)
+mesh = P4estMeshCubedSphere2D(5, 1.0, polydeg = polydeg,
+                              initial_refinement_level = 0,
+                              element_local_mapping = element_local_mapping)
 
 # A semidiscretization collects data structures and functions for the spatial discretization
-semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver)
+semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver,
+                                    source_terms = source_terms_convert_to_linear_advection)
 
 ###############################################################################
 # ODE solvers, callbacks etc.
@@ -117,12 +90,6 @@ semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver)
 tspan = (0.0, pi)
 ode = semidiscretize(semi, tspan)
 
-# Create custom discretization that runs with the custom RHS
-ode_semi_custom = ODEProblem(rhs_semi_custom!,
-                             ode.u0,
-                             ode.tspan,
-                             semi)
-
 # At the beginning of the main loop, the SummaryCallback prints a summary of the simulation setup
 # and resets the timers
 summary_callback = SummaryCallback()
@@ -148,7 +115,7 @@ callbacks = CallbackSet(summary_callback, analysis_callback, save_solution,
 # run the simulation
 
 # OrdinaryDiffEq's `solve` method evolves the solution in time and executes the passed callbacks
-sol = solve(ode_semi_custom, CarpenterKennedy2N54(williamson_condition = false),
+sol = solve(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);
 

examples/elixir_shallowwater_cubed_sphere_shell_EC_correction.jl

@@ -0,0 +1,125 @@
+
+using OrdinaryDiffEq
+using Trixi
+using TrixiAtmo
+###############################################################################
+# Entropy conservation for the spherical shallow water equations in Cartesian
+# form obtained through an entropy correction term
+
+equations = ShallowWaterEquations3D(gravity_constant = 9.81)
+
+# Create DG solver with polynomial degree = 3 and Wintemeyer et al.'s flux as surface flux
+polydeg = 3
+volume_flux = flux_fjordholm_etal #flux_wintermeyer_etal
+surface_flux = flux_fjordholm_etal #flux_wintermeyer_etal  #flux_lax_friedrichs
+solver = DGSEM(polydeg = polydeg,
+               surface_flux = surface_flux,
+               volume_integral = VolumeIntegralFluxDifferencing(volume_flux))
+
+# Initial condition for a Gaussian density profile with constant pressure
+# and the velocity of a rotating solid body
+function initial_condition_advection_sphere(x, t, equations::ShallowWaterEquations3D)
+    # Gaussian density
+    rho = 1.0 + exp(-20 * (x[1]^2 + x[3]^2))
+
+    # Spherical coordinates for the point x
+    if sign(x[2]) == 0.0
+        signy = 1.0
+    else
+        signy = sign(x[2])
+    end
+    # Co-latitude
+    colat = acos(x[3] / sqrt(x[1]^2 + x[2]^2 + x[3]^2))
+    # Latitude (auxiliary variable)
+    lat = -colat + 0.5 * pi
+    # Longitude
+    r_xy = sqrt(x[1]^2 + x[2]^2)
+    if r_xy == 0.0
+        phi = pi / 2
+    else
+        phi = signy * acos(x[1] / r_xy)
+    end
+
+    # Compute the velocity of a rotating solid body
+    # (alpha is the angle between the rotation axis and the polar axis of the spherical coordinate system)
+    v0 = 1.0 # Velocity at the "equator"
+    alpha = 0.0 #pi / 4
+    v_long = v0 * (cos(lat) * cos(alpha) + sin(lat) * cos(phi) * sin(alpha))
+    v_lat = -v0 * sin(phi) * sin(alpha)
+
+    # Transform to Cartesian coordinate system
+    v1 = -cos(colat) * cos(phi) * v_lat - sin(phi) * v_long
+    v2 = -cos(colat) * sin(phi) * v_lat + cos(phi) * v_long
+    v3 = sin(colat) * v_lat
+
+    return prim2cons(SVector(rho, v1, v2, v3, 0), equations)
+end
+
+# Source term function to apply the entropy correction term and the Lagrange multiplier to the semi-discretization.
+# The vector contravariant_normal_vector contains the normal contravariant vectors scaled with the inverse Jacobian.
+# The Lagrange multiplier is applied with the vector x, which contains the exact normal direction to the 2D manifold.
+function source_terms_entropy_correction(u, du, x, t,
+                                         equations::ShallowWaterEquations3D,
+                                         contravariant_normal_vector)
+
+    # Entropy correction term
+    s2 = -contravariant_normal_vector[1] * equations.gravity * u[1]^2
+    s3 = -contravariant_normal_vector[2] * equations.gravity * u[1]^2
+    s4 = -contravariant_normal_vector[3] * equations.gravity * u[1]^2
+
+    new_du = SVector(du[1], du[2] + s2, du[3] + s3, du[4] + s4, du[5])
+
+    # Apply Lagrange multipliers using the exact normal vector to the 2D manifold (x)
+    s = source_terms_lagrange_multiplier(u, new_du, x, t, equations, x)
+
+    return SVector(s[1], s[2] + s2, s[3] + s3, s[4] + s4, s[5])
+end
+
+initial_condition = initial_condition_advection_sphere
+
+mesh = P4estMeshCubedSphere2D(5, 2.0, polydeg = polydeg,
+                              initial_refinement_level = 0)
+
+# A semidiscretization collects data structures and functions for the spatial discretization
+semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver,
+                                    metric_terms = MetricTermsInvariantCurl(),
+                                    source_terms = source_terms_entropy_correction)
+
+###############################################################################
+# ODE solvers, callbacks etc.
+
+# Create ODE problem with time span from 0.0 to π
+tspan = (0.0, pi)
+ode = semidiscretize(semi, tspan)
+
+# At the beginning of the main loop, the SummaryCallback prints a summary of the simulation setup
+# and resets the timers
+summary_callback = SummaryCallback()
+
+# The AnalysisCallback allows to analyse the solution in regular intervals and prints the results
+analysis_callback = AnalysisCallback(semi, interval = 10,
+                                     save_analysis = true,
+                                     extra_analysis_errors = (:conservation_error,),
+                                     extra_analysis_integrals = (waterheight, energy_total))
+
+# The SaveSolutionCallback allows to save the solution to a file in regular intervals
+save_solution = SaveSolutionCallback(interval = 10,
+                                     solution_variables = cons2prim)
+
+# The StepsizeCallback handles the re-calculation of the maximum Δt after each time step
+stepsize_callback = StepsizeCallback(cfl = 0.7)
+
+# Create a CallbackSet to collect all callbacks such that they can be passed to the ODE solver
+callbacks = CallbackSet(summary_callback, analysis_callback, save_solution,
+                        stepsize_callback)
+
+###############################################################################
+# run the simulation
+
+# OrdinaryDiffEq's `solve` method evolves the solution in time and executes the passed callbacks
+sol = solve(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);
+
+# Print the timer summary
+summary_callback()

examples/elixir_shallowwater_cubed_sphere_shell_EC_projection.jl

@@ -0,0 +1,120 @@
+
+using OrdinaryDiffEq
+using Trixi
+using TrixiAtmo
+
+###############################################################################
+# Entropy conservation for the spherical shallow water equations in Cartesian
+# form obtained through the projection of the momentum onto the divergence-free
+# tangential contravariant vectors
+
+equations = ShallowWaterEquations3D(gravity_constant = 9.81)
+
+# Create DG solver with polynomial degree = 3 and Wintemeyer et al.'s flux as surface flux
+polydeg = 3
+volume_flux = flux_wintermeyer_etal # flux_fjordholm_etal 
+surface_flux = flux_wintermeyer_etal # flux_fjordholm_etal #flux_lax_friedrichs
+solver = DGSEM(polydeg = polydeg,
+               surface_flux = surface_flux,
+               volume_integral = VolumeIntegralFluxDifferencing(volume_flux))
+
+# Initial condition for a Gaussian density profile with constant pressure
+# and the velocity of a rotating solid body
+function initial_condition_advection_sphere(x, t, equations::ShallowWaterEquations3D)
+    # Gaussian density
+    rho = 1.0 + exp(-20 * (x[1]^2 + x[3]^2))
+
+    # Spherical coordinates for the point x
+    if sign(x[2]) == 0.0
+        signy = 1.0
+    else
+        signy = sign(x[2])
+    end
+    # Co-latitude
+    colat = acos(x[3] / sqrt(x[1]^2 + x[2]^2 + x[3]^2))
+    # Latitude (auxiliary variable)
+    lat = -colat + 0.5 * pi
+    # Longitude
+    r_xy = sqrt(x[1]^2 + x[2]^2)
+    if r_xy == 0.0
+        phi = pi / 2
+    else
+        phi = signy * acos(x[1] / r_xy)
+    end
+
+    # Compute the velocity of a rotating solid body
+    # (alpha is the angle between the rotation axis and the polar axis of the spherical coordinate system)
+    v0 = 1.0 # Velocity at the "equator"
+    alpha = 0.0 #pi / 4
+    v_long = v0 * (cos(lat) * cos(alpha) + sin(lat) * cos(phi) * sin(alpha))
+    v_lat = -v0 * sin(phi) * sin(alpha)
+
+    # Transform to Cartesian coordinate system
+    v1 = -cos(colat) * cos(phi) * v_lat - sin(phi) * v_long
+    v2 = -cos(colat) * sin(phi) * v_lat + cos(phi) * v_long
+    v3 = sin(colat) * v_lat
+
+    return prim2cons(SVector(rho, v1, v2, v3, 0), equations)
+end
+
+initial_condition = initial_condition_advection_sphere
+
+mesh = P4estMeshCubedSphere2D(5, 2.0, polydeg = polydeg,
+                              initial_refinement_level = 0)
+
+# A semidiscretization collects data structures and functions for the spatial discretization
+semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver,
+                                    metric_terms = MetricTermsInvariantCurl(),
+                                    source_terms = source_terms_lagrange_multiplier)
+
+###############################################################################
+# ODE solvers, callbacks etc.
+
+# Create ODE problem with time span from 0.0 to π
+tspan = (0.0, pi)
+ode = semidiscretize(semi, tspan)
+
+# Clean the initial condition
+for element in eachelement(solver, semi.cache)
+    for j in eachnode(solver), i in eachnode(solver)
+        u0 = Trixi.wrap_array(ode.u0, semi)
+
+        contravariant_normal_vector = Trixi.get_contravariant_vector(3,
+                                                                     semi.cache.elements.contravariant_vectors,
+                                                                     i, j, element)
+        clean_solution_lagrange_multiplier!(u0[:, i, j, element], equations,
+                                            contravariant_normal_vector)
+    end
+end
+
+# At the beginning of the main loop, the SummaryCallback prints a summary of the simulation setup
+# and resets the timers
+summary_callback = SummaryCallback()
+
+# The AnalysisCallback allows to analyse the solution in regular intervals and prints the results
+analysis_callback = AnalysisCallback(semi, interval = 10,
+                                     save_analysis = true,
+                                     extra_analysis_errors = (:conservation_error,),
+                                     extra_analysis_integrals = (waterheight, energy_total))
+
+# The SaveSolutionCallback allows to save the solution to a file in regular intervals
+save_solution = SaveSolutionCallback(interval = 10,
+                                     solution_variables = cons2prim)
+
+# The StepsizeCallback handles the re-calculation of the maximum Δt after each time step
+stepsize_callback = StepsizeCallback(cfl = 0.7)
+
+# Create a CallbackSet to collect all callbacks such that they can be passed to the ODE solver
+callbacks = CallbackSet(summary_callback, analysis_callback, save_solution,
+                        stepsize_callback)
+
+###############################################################################
+# run the simulation
+
+# OrdinaryDiffEq's `solve` method evolves the solution in time and executes the passed callbacks
+sol = solve(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);
+
+# Print the timer summary
+summary_callback()