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trixi-framework
GitHub Repository: trixi-framework/Trixi.jl
 Path: blob/main/examples/tree_3d_dgsem/elixir_hypdiff_lax_friedrichs.jl
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using Trixi

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###############################################################################

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# semidiscretization of the hyperbolic diffusion equations

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equations = HyperbolicDiffusionEquations3D()

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function initial_condition_poisson_periodic(x, t, equations::HyperbolicDiffusionEquations3D)

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    # elliptic equation: -νΔϕ = f

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    # depending on initial constant state, c, for phi this converges to the solution ϕ + c

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    if iszero(t)

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        phi = 0.0

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        q1 = 0.0

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        q2 = 0.0

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        q3 = 0.0

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    else

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        phi = sin(2 * pi * x[1]) * sin(2 * pi * x[2]) * sin(2 * pi * x[3])

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        q1 = 2 * pi * cos(2 * pi * x[1]) * sin(2 * pi * x[2]) * sin(2 * pi * x[3])

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        q2 = 2 * pi * sin(2 * pi * x[1]) * cos(2 * pi * x[2]) * sin(2 * pi * x[3])

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        q3 = 2 * pi * sin(2 * pi * x[1]) * sin(2 * pi * x[2]) * cos(2 * pi * x[3])

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    end

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    return SVector(phi, q1, q2, q3)

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end

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initial_condition = initial_condition_poisson_periodic

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@inline function source_terms_poisson_periodic(u, x, t,

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                                               equations::HyperbolicDiffusionEquations3D)

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    # elliptic equation: -νΔϕ = f

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    # analytical solution: phi = sin(2πx)*sin(2πy) and f = -8νπ^2 sin(2πx)*sin(2πy)

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    @unpack inv_Tr = equations

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    C = -12 * equations.nu * pi^2

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    x1, x2, x3 = x

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    tmp1 = sinpi(2 * x1)

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    tmp2 = sinpi(2 * x2)

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    tmp3 = sinpi(2 * x3)

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    du1 = -C * tmp1 * tmp2 * tmp3

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    du2 = -inv_Tr * u[2]

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    du3 = -inv_Tr * u[3]

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    du4 = -inv_Tr * u[4]

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    return SVector(du1, du2, du3, du4)

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end

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solver = DGSEM(polydeg = 3, surface_flux = flux_lax_friedrichs)

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coordinates_min = (0.0, 0.0, 0.0)

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coordinates_max = (1.0, 1.0, 1.0)

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mesh = TreeMesh(coordinates_min, coordinates_max,

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                initial_refinement_level = 3,

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                n_cells_max = 30_000)

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semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver,

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                                    source_terms = source_terms_poisson_periodic)

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###############################################################################

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# ODE solvers, callbacks etc.

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tspan = (0.0, 2.0)

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ode = semidiscretize(semi, tspan)

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summary_callback = SummaryCallback()

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resid_tol = 5.0e-12

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steady_state_callback = SteadyStateCallback(abstol = resid_tol, reltol = 0.0)

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analysis_interval = 200

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analysis_callback = AnalysisCallback(semi, interval = analysis_interval,

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                                     extra_analysis_integrals = (entropy, energy_total))

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alive_callback = AliveCallback(analysis_interval = analysis_interval)

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save_solution = SaveSolutionCallback(interval = 100,

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                                     save_initial_solution = true,

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                                     save_final_solution = true,

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                                     solution_variables = cons2prim)

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stepsize_callback = StepsizeCallback(cfl = 2.4)

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callbacks = CallbackSet(summary_callback, steady_state_callback,

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                        analysis_callback, alive_callback,

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                        save_solution,

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                        stepsize_callback)

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###############################################################################

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# run the simulation

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sol = Trixi.solve(ode, Trixi.HypDiffN3Erk3Sstar52();

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                  dt = 1.0, # solve needs some value here but it will be overwritten by the stepsize_callback

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                  ode_default_options()..., callback = callbacks);

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