elixir_mhd_ec.jl

using OrdinaryDiffEqLowStorageRK
using Trixi

###############################################################################
# semidiscretization of the compressible ideal GLM-MHD equations

equations = IdealGlmMhdEquations2D(1.4)

function initial_condition_shifted_weak_blast_wave(x, t, equations::IdealGlmMhdEquations2D)
    # Adapted MHD version of the weak blast wave from Hennemann & Gassner JCP paper 2020 (Sec. 6.3)
    # Same discontinuity in the velocities but with magnetic fields
    # Set up polar coordinates
    inicenter = (1.5, 1.5)
    x_norm = x[1] - inicenter[1]
    y_norm = x[2] - inicenter[2]
    r = sqrt(x_norm^2 + y_norm^2)
    phi = atan(y_norm, x_norm)

    # Calculate primitive variables
    rho = r > 0.5 ? 1.0 : 1.1691
    v1 = r > 0.5 ? 0.0 : 0.1882 * cos(phi)
    v2 = r > 0.5 ? 0.0 : 0.1882 * sin(phi)
    p = r > 0.5 ? 1.0 : 1.245

    return prim2cons(SVector(rho, v1, v2, 0.0, p, 1.0, 1.0, 1.0, 0.0), equations)
end

initial_condition = initial_condition_shifted_weak_blast_wave

# Get the DG approximation space
volume_flux = (flux_hindenlang_gassner, flux_nonconservative_powell)
solver = DGSEM(polydeg = 5,
               surface_flux = (flux_hindenlang_gassner, flux_nonconservative_powell),
               volume_integral = VolumeIntegralFluxDifferencing(volume_flux))

# Get the curved quad mesh from a mapping function
# Mapping as described in https://arxiv.org/abs/2012.12040, but reduced to 2D
function mapping(xi_, eta_)
    # Transform input variables between -1 and 1 onto [0,3]
    xi = 1.5 * xi_ + 1.5
    eta = 1.5 * eta_ + 1.5

    y = eta + 3 / 8 * (cos(1.5 * pi * (2 * xi - 3) / 3) *
                       cos(0.5 * pi * (2 * eta - 3) / 3))

    x = xi + 3 / 8 * (cos(0.5 * pi * (2 * xi - 3) / 3) *
                      cos(2 * pi * (2 * y - 3) / 3))

    return SVector(x, y)
end

# Create curved mesh with 8 x 8 elements
cells_per_dimension = (8, 8)
mesh = StructuredMesh(cells_per_dimension, mapping, periodicity = true)

# create the semi discretization object
semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver;
                                    boundary_conditions = boundary_condition_periodic)

###############################################################################
# ODE solvers, callbacks etc.

tspan = (0.0, 2.0)
ode = semidiscretize(semi, tspan)

summary_callback = SummaryCallback()

analysis_interval = 100
analysis_callback = AnalysisCallback(semi, interval = analysis_interval,
                                     save_analysis = false,
                                     extra_analysis_integrals = (entropy, energy_total,
                                                                 energy_kinetic,
                                                                 energy_internal,
                                                                 energy_magnetic,
                                                                 cross_helicity))

alive_callback = AliveCallback(analysis_interval = analysis_interval)

save_solution = SaveSolutionCallback(interval = 100,
                                     save_initial_solution = true,
                                     save_final_solution = true,
                                     solution_variables = cons2prim)
cfl = 1.0
stepsize_callback = StepsizeCallback(cfl = cfl)

glm_speed_callback = GlmSpeedCallback(glm_scale = 0.5, cfl = cfl)

callbacks = CallbackSet(summary_callback,
                        analysis_callback,
                        alive_callback,
                        save_solution,
                        stepsize_callback,
                        glm_speed_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
            ode_default_options()..., callback = callbacks);