Add Legendre-Gauss basis for DGSEM and implement solver support for 1D/2D TreeMesh

NEWS.md

@@ -26,6 +26,7 @@ This enables in particular adaptive mesh refinement for that solver-mesh combina
 - The second-order subcell volume integral is no longer limited to reconstruction in primitive variables.
   Instead, it is possible to reconstruct in custom variables, if functions `cons2recon` and `recon2cons` are provided to
   `VolumeIntegralPureLGLFiniteVolumeO2` and `VolumeIntegralShockCapturingRRG`([#2817]).
+- Add Legendre-Gauss basis for DGSEM and implement solver (`VolumeIntegralWeakForm` and `SurfaceIntegralWeakForm` only) support for conforming 1D & 2D `TreeMesh`es ([#1965]).
 
 ## Changes when updating to v0.15 from v0.14.x
 

examples/tree_1d_dgsem/elixir_advection_gauss_legendre.jl

@@ -0,0 +1,42 @@
+using OrdinaryDiffEqLowStorageRK
+using Trixi
+
+###############################################################################
+# semidiscretization of the linear advection equation
+
+advection_velocity = 1.0
+equations = LinearScalarAdvectionEquation1D(advection_velocity)
+
+basis = GaussLegendreBasis(3)
+solver = DGSEM(basis, flux_godunov)
+
+coordinates_min = -1.0
+coordinates_max = 1.0
+
+mesh = TreeMesh(coordinates_min, coordinates_max,
+                initial_refinement_level = 4,
+                n_cells_max = 30_000, periodicity = true)
+
+semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition_convergence_test,
+                                    solver;
+                                    boundary_conditions = boundary_condition_periodic)
+
+###############################################################################
+# ODE solvers, callbacks etc.
+
+ode = semidiscretize(semi, (0.0, 1.0))
+
+summary_callback = SummaryCallback()
+
+analysis_callback = AnalysisCallback(semi, interval = 100)
+
+stepsize_callback = StepsizeCallback(cfl = 0.8)
+
+callbacks = CallbackSet(summary_callback, analysis_callback,
+                        stepsize_callback)
+
+###############################################################################
+# run the simulation
+
+sol = solve(ode, CarpenterKennedy2N54(williamson_condition = false); dt = 1.0,
+            ode_default_options()..., callback = callbacks);

examples/tree_2d_dgsem/elixir_euler_convergence_gauss.jl

@@ -0,0 +1,52 @@
+using OrdinaryDiffEqLowStorageRK
+using Trixi
+
+###############################################################################
+# semidiscretization of the compressible Euler equations
+
+equations = CompressibleEulerEquations2D(1.4)
+
+initial_condition = initial_condition_convergence_test
+
+surface_flux = flux_lax_friedrichs
+
+polydeg = 3
+basis = GaussLegendreBasis(polydeg)
+solver = DGSEM(polydeg = 3, basis = basis, surface_flux = surface_flux)
+
+coordinates_min = (0.0, 0.0)
+coordinates_max = (2.0, 2.0)
+mesh = TreeMesh(coordinates_min, coordinates_max,
+                initial_refinement_level = 2,
+                n_cells_max = 100_000,
+                periodicity = true)
+
+semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver,
+                                    source_terms = source_terms_convergence_test;
+                                    boundary_conditions = boundary_condition_periodic)
+
+###############################################################################
+# ODE solvers, callbacks etc.
+
+tspan = (0.0, 2.0)
+ode = semidiscretize(semi, tspan)
+
+summary_callback = SummaryCallback()
+
+analysis_interval = 1000
+analysis_callback = AnalysisCallback(semi, interval = analysis_interval)
+
+alive_callback = AliveCallback(analysis_interval = analysis_interval)
+
+stepsize_callback = StepsizeCallback(cfl = 0.9)
+
+callbacks = CallbackSet(summary_callback,
+                        analysis_callback, alive_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
+            save_everystep = false, callback = callbacks);

src/Trixi.jl

@@ -263,7 +263,7 @@ export TreeMesh, StructuredMesh, StructuredMeshView, UnstructuredMesh2D, P4estMe
        P4estMeshView, P4estMeshCubedSphere, T8codeMesh
 
 export DG,
-       DGSEM, LobattoLegendreBasis,
+       DGSEM, LobattoLegendreBasis, GaussLegendreBasis,
        FDSBP,
        VolumeIntegralWeakForm, VolumeIntegralStrongForm,
        VolumeIntegralFluxDifferencing,

src/auxiliary/precompile.jl

@@ -315,7 +315,7 @@ function _precompile_manual_()
         @assert Base.precompile(Tuple{typeof(Trixi.polynomial_interpolation_matrix),
                                       Vector{RealT}, Vector{RealT}})
         @assert Base.precompile(Tuple{typeof(Trixi.barycentric_weights), Vector{RealT}})
-        @assert Base.precompile(Tuple{typeof(Trixi.calc_Lhat), RealT, Vector{RealT},
+        @assert Base.precompile(Tuple{typeof(Trixi.calc_L), RealT, Vector{RealT},
                                       Vector{RealT}})
         @assert Base.precompile(Tuple{typeof(Trixi.lagrange_interpolating_polynomials),
                                       RealT, Vector{RealT}, Vector{RealT}})