Implement the Cubed Sphere as P4estMesh (#689)

* Implement cubed sphere connectivity

* Rebuild cubed sphere connectivity with only right-handed elements

* Fix cubed sphere with more cells in x and y direction

* Make #layers of the sphere customizable

* Compute a non-constant IC on the sphere

* Add stupid comment that I spent way too much time on

* Improve documentation

* Add comment

* Implement suggestions

* Add cubed sphere elixir to tests

examples/p4est_3d_dgsem/elixir_advection_cubed_sphere.jl

@@ -0,0 +1,65 @@
+
+using OrdinaryDiffEq
+using Trixi
+
+###############################################################################
+# semidiscretization of the linear advection equation
+
+advectionvelocity = (1.0, 1.0, 1.0)
+equations = LinearScalarAdvectionEquation3D(advectionvelocity)
+
+# Create DG solver with polynomial degree = 3 and (local) Lax-Friedrichs/Rusanov flux as surface flux
+solver = DGSEM(polydeg=3, surface_flux=flux_lax_friedrichs)
+
+initial_condition = initial_condition_convergence_test
+
+boundary_condition = BoundaryConditionDirichlet(initial_condition)
+boundary_conditions = Dict(
+  :inside  => boundary_condition,
+  :outside => boundary_condition,
+)
+
+mesh = Trixi.P4estMeshCubedSphere(3, 2, 0.5, 0.5,
+                                  polydeg=3, initial_refinement_level=0)
+
+# A semidiscretization collects data structures and functions for the spatial discretization
+semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver, boundary_conditions=boundary_conditions)
+
+###############################################################################
+# ODE solvers, callbacks etc.
+
+# Create ODE problem with time span from 0.0 to 1.0
+ode = semidiscretize(semi, (0.0, 1.0));
+
+# 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=100)
+
+# The SaveRestartCallback allows to save a file from which a Trixi simulation can be restarted
+save_restart = SaveRestartCallback(interval=100,
+                                   save_final_restart=true)
+
+# The SaveSolutionCallback allows to save the solution to a file in regular intervals
+save_solution = SaveSolutionCallback(interval=100,
+                                     solution_variables=cons2prim)
+
+# The StepsizeCallback handles the re-calculcation of the maximum Δt after each time step
+stepsize_callback = StepsizeCallback(cfl=1.2)
+
+# Create a CallbackSet to collect all callbacks such that they can be passed to the ODE solver
+callbacks = CallbackSet(summary_callback, analysis_callback, save_restart, 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()