elixir_advection_diffusion_nonperiodic.jl

using OrdinaryDiffEqLowStorageRK
using Trixi

###############################################################################
# semidiscretization of the linear advection-diffusion equation

diffusivity() = 5.0e-2
advection_velocity = (1.0, 0.0)
equations = LinearScalarAdvectionEquation2D(advection_velocity)
equations_parabolic = LaplaceDiffusion2D(diffusivity(), equations)

# 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)

coordinates_min = (-1.0, -0.5) # minimum coordinates (min(x), min(y))
coordinates_max = (0.0, 0.5) # maximum coordinates (max(x), max(y))

# Create a uniformly refined mesh with periodic boundaries
mesh = TreeMesh(coordinates_min, coordinates_max,
                initial_refinement_level = 4,
                periodicity = false,
                n_cells_max = 30_000) # set maximum capacity of tree data structure

# Example setup taken from
# - Truman Ellis, Jesse Chan, and Leszek Demkowicz (2016).
#   Robust DPG methods for transient convection-diffusion.
#   In: Building bridges: connections and challenges in modern approaches
#   to numerical partial differential equations.
#   [DOI](https://doi.org/10.1007/978-3-319-41640-3_6).
function initial_condition_eriksson_johnson(x, t, equations)
    RealT = eltype(x)
    l = 4
    epsilon = diffusivity() # TODO: this requires epsilon < .6 due to sqrt
    lambda_1 = (-1 + sqrt(1 - 4 * epsilon * l)) / (-2 * epsilon)
    lambda_2 = (-1 - sqrt(1 - 4 * epsilon * l)) / (-2 * epsilon)
    r1 = (1 + sqrt(1 + 4 * convert(RealT, pi)^2 * epsilon^2)) / (2 * epsilon)
    s1 = (1 - sqrt(1 + 4 * convert(RealT, pi)^2 * epsilon^2)) / (2 * epsilon)
    u = exp(-l * t) * (exp(lambda_1 * x[1]) - exp(lambda_2 * x[1])) +
        cospi(x[2]) * (exp(s1 * x[1]) - exp(r1 * x[1])) / (exp(-s1) - exp(-r1))
    return SVector{1}(u)
end
initial_condition = initial_condition_eriksson_johnson

boundary_conditions = (; x_neg = BoundaryConditionDirichlet(initial_condition),
                       y_neg = BoundaryConditionDirichlet(initial_condition),
                       y_pos = BoundaryConditionDirichlet(initial_condition),
                       x_pos = boundary_condition_do_nothing)

boundary_conditions_parabolic = BoundaryConditionDirichlet(initial_condition)

# A semidiscretization collects data structures and functions for the spatial discretization
semi = SemidiscretizationHyperbolicParabolic(mesh,
                                             (equations, equations_parabolic),
                                             initial_condition, solver;
                                             solver_parabolic = ViscousFormulationBassiRebay1(),
                                             boundary_conditions = (boundary_conditions,
                                                                    boundary_conditions_parabolic))

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

# Create ODE problem with time span `tspan`
tspan = (0.0, 1.5)
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_interval = 100
analysis_callback = AnalysisCallback(semi, interval = analysis_interval)

# The AliveCallback prints short status information in regular intervals
alive_callback = AliveCallback(analysis_interval = analysis_interval)

# Create a CallbackSet to collect all callbacks such that they can be passed to the ODE solver
callbacks = CallbackSet(summary_callback, analysis_callback, alive_callback)

###############################################################################
# run the simulation

# OrdinaryDiffEq's `solve` method evolves the solution in time and executes the passed callbacks
time_int_tol = 1.0e-11
sol = solve(ode, RDPK3SpFSAL49(); abstol = time_int_tol, reltol = time_int_tol,
            ode_default_options()..., callback = callbacks)