12: Parabolic terms
Experimental support for parabolic diffusion terms is available in Trixi.jl. This demo illustrates parabolic terms for the advection-diffusion equation.
using OrdinaryDiffEq
using TrixiSplitting a system into hyperbolic and parabolic parts.
For a mixed hyperbolic-parabolic system, we represent the hyperbolic and parabolic parts of the system separately. We first define the hyperbolic (advection) part of the advection-diffusion equation.
advection_velocity = (1.5, 1.0)
equations_hyperbolic = LinearScalarAdvectionEquation2D(advection_velocity);Next, we define the parabolic diffusion term. The constructor requires knowledge of equations_hyperbolic to be passed in because the LaplaceDiffusion2D applies diffusion to every variable of the hyperbolic system.
diffusivity = 5.0e-2
equations_parabolic = LaplaceDiffusion2D(diffusivity, equations_hyperbolic);Boundary conditions
As with the equations, we define boundary conditions separately for the hyperbolic and parabolic part of the system. For this example, we impose inflow BCs for the hyperbolic system (no condition is imposed on the outflow), and we impose Dirichlet boundary conditions for the parabolic equations. Both BoundaryConditionDirichlet and BoundaryConditionNeumann are defined for LaplaceDiffusion2D.
The hyperbolic and parabolic boundary conditions are assumed to be consistent with each other.
boundary_condition_zero_dirichlet = BoundaryConditionDirichlet((x, t, equations) -> SVector(0.0))
boundary_conditions_hyperbolic = (; x_neg = BoundaryConditionDirichlet((x, t, equations) -> SVector(1 + 0.5 * x[2])),
y_neg = boundary_condition_zero_dirichlet,
y_pos = boundary_condition_do_nothing,
x_pos = boundary_condition_do_nothing)
boundary_conditions_parabolic = (; x_neg = BoundaryConditionDirichlet((x, t, equations) -> SVector(1 + 0.5 * x[2])),
y_neg = boundary_condition_zero_dirichlet,
y_pos = boundary_condition_zero_dirichlet,
x_pos = boundary_condition_zero_dirichlet);Defining the solver and mesh
The process of creating the DG solver and mesh is the same as for a purely hyperbolic system of equations.
solver = DGSEM(polydeg=3, surface_flux=flux_lax_friedrichs)
coordinates_min = (-1.0, -1.0) # minimum coordinates (min(x), min(y))
coordinates_max = ( 1.0, 1.0) # maximum coordinates (max(x), max(y))
mesh = TreeMesh(coordinates_min, coordinates_max,
initial_refinement_level=4,
periodicity=false, n_cells_max=30_000) # set maximum capacity of tree data structure
initial_condition = (x, t, equations) -> SVector(0.0);Semidiscretizing and solving
To semidiscretize a hyperbolic-parabolic system, we create a SemidiscretizationHyperbolicParabolic. This differs from a SemidiscretizationHyperbolic in that we pass in a Tuple containing both the hyperbolic and parabolic equation, as well as a Tuple containing the hyperbolic and parabolic boundary conditions.
semi = SemidiscretizationHyperbolicParabolic(mesh,
(equations_hyperbolic, equations_parabolic),
initial_condition, solver;
boundary_conditions=(boundary_conditions_hyperbolic,
boundary_conditions_parabolic))┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ SemidiscretizationHyperbolicParabolic │
│ ═════════════════════════════════════ │
│ #spatial dimensions: ………………………… 2 │
│ mesh: ………………………………………………………………… TreeMesh{2, Trixi.SerialTree{2}} with length 341 │
│ hyperbolic equations: ……………………… LinearScalarAdvectionEquation2D │
│ parabolic equations: ………………………… LaplaceDiffusion2D │
│ initial condition: ……………………………… #7 │
│ source terms: …………………………………………… nothing │
│ solver: …………………………………………………………… DG │
│ parabolic solver: ………………………………… ViscousFormulationBassiRebay1 │
│ total #DOFs per field: …………………… 4096 │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘The rest of the code is identical to the hyperbolic case. We create a system of ODEs through semidiscretize, defining callbacks, and then passing the system to OrdinaryDiffEq.jl.
tspan = (0.0, 1.5)
ode = semidiscretize(semi, tspan)
callbacks = CallbackSet(SummaryCallback())
time_int_tol = 1.0e-6
sol = solve(ode, RDPK3SpFSAL49(); abstol=time_int_tol, reltol=time_int_tol,
ode_default_options()..., callback=callbacks);
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┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ SemidiscretizationHyperbolicParabolic │
│ ═════════════════════════════════════ │
│ #spatial dimensions: ………………………… 2 │
│ mesh: ………………………………………………………………… TreeMesh{2, Trixi.SerialTree{2}} with length 341 │
│ hyperbolic equations: ……………………… LinearScalarAdvectionEquation2D │
│ parabolic equations: ………………………… LaplaceDiffusion2D │
│ initial condition: ……………………………… #7 │
│ source terms: …………………………………………… nothing │
│ solver: …………………………………………………………… DG │
│ parabolic solver: ………………………………… ViscousFormulationBassiRebay1 │
│ total #DOFs per field: …………………… 4096 │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘
┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ TreeMesh{2, Trixi.SerialTree{2}} │
│ ════════════════════════════════ │
│ center: …………………………………………………………… [0.0, 0.0] │
│ length: …………………………………………………………… 2.0 │
│ periodicity: ……………………………………………… (false, false) │
│ current #cells: ……………………………………… 341 │
│ #leaf-cells: ……………………………………………… 256 │
│ maximum #cells: ……………………………………… 30000 │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘
┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ LinearScalarAdvectionEquation2D │
│ ═══════════════════════════════ │
│ #variables: ………………………………………………… 1 │
│ │ variable 1: …………………………………………… scalar │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘
┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ DG{Float64} │
│ ═══════════ │
│ basis: ……………………………………………………………… LobattoLegendreBasis{Float64}(polydeg=3) │
│ mortar: …………………………………………………………… LobattoLegendreMortarL2{Float64}(polydeg=3) │
│ surface integral: ………………………………… SurfaceIntegralWeakForm │
│ │ surface flux: ……………………………………… FluxLaxFriedrichs(max_abs_speed_naive) │
│ volume integral: …………………………………… VolumeIntegralWeakForm │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘
┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ Time integration │
│ ════════════════ │
│ Start time: ………………………………………………… 0.0 │
│ Final time: ………………………………………………… 1.5 │
│ time integrator: …………………………………… RDPK3SpFSAL49 │
│ adaptive: ……………………………………………………… true │
│ abstol: …………………………………………………………… 1.0e-6 │
│ reltol: …………………………………………………………… 1.0e-6 │
│ controller: ………………………………………………… PIDController(beta=[0.38, -0.18,…iter=default_dt_factor_limiter) │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘
┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ Environment information │
│ ═══════════════════════ │
│ #threads: ……………………………………………………… 1 │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘We can now visualize the solution, which develops a boundary layer at the outflow boundaries.
using Plots
plot(sol)
Package versions
These results were obtained using the following versions.
using InteractiveUtils
versioninfo()
using Pkg
Pkg.status(["Trixi", "OrdinaryDiffEq", "Plots"],
mode=PKGMODE_MANIFEST)Julia Version 1.9.4
Commit 8e5136fa297 (2023-11-14 08:46 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 4 × AMD EPYC 7763 64-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-14.0.6 (ORCJIT, znver3)
Threads: 1 on 4 virtual cores
Environment:
JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
Status `~/work/Trixi.jl/Trixi.jl/docs/Manifest.toml`
⌅ [1dea7af3] OrdinaryDiffEq v6.66.0
[91a5bcdd] Plots v1.40.4
[a7f1ee26] Trixi v0.7.8 `~/work/Trixi.jl/Trixi.jl`
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`This page was generated using Literate.jl.