13: Parabolic terms
Experimental support for parabolic diffusion terms is available in Trixi.jl. This demo illustrates parabolic terms for the advection-diffusion equation.
using OrdinaryDiffEqLowStorageRKusing 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-2equations_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)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, Float64}} with length 341 │
│ hyperbolic equations: …………………………………… LinearScalarAdvectionEquation2D │
│ parabolic equations: ……………………………………… LaplaceDiffusion2D │
│ initial condition: …………………………………………… #7 │
│ source terms: ………………………………………………………… nothing │
│ source terms parabolic: ……………………………… nothing │
│ solver: ………………………………………………………………………… DG │
│ parabolic solver: ……………………………………………… ParabolicFormulationBassiRebay1 │
│ 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-6sol = 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, Float64}} with length 341 │
│ hyperbolic equations: …………………………………… LinearScalarAdvectionEquation2D │
│ parabolic equations: ……………………………………… LaplaceDiffusion2D │
│ initial condition: …………………………………………… #7 │
│ source terms: ………………………………………………………… nothing │
│ source terms parabolic: ……………………………… nothing │
│ solver: ………………………………………………………………………… DG │
│ parabolic solver: ……………………………………………… ParabolicFormulationBassiRebay1 │
│ total #DOFs per field: ………………………………… 4096 │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘
┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ TreeMesh{2, Trixi.SerialTree{2, Float64}} │
│ ═════════════════════════════════════════ │
│ center: ………………………………………………………………………… [0.0, 0.0] │
│ length: ………………………………………………………………………… 2.0 │
│ periodicity: …………………………………………………………… (false, false) │
│ current #cells: …………………………………………………… 341 │
│ #leaf-cells: …………………………………………………………… 256 │
│ current capacity: ……………………………………………… 341 │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘
┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ 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) │
│ 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.…er=default_dt_factor_limiter) │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘
┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ Environment information │
│ ═══════════════════════ │
│ #threads: …………………………………………………………………… 1 │
│ threading backend: …………………………………………… polyester │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘
Trixi.jl
───────────────────────────────────────────────────────────────────────────────────────────
Time Allocations
────────────────────── ────────────────────────
Tot / % measured: 73.6ms / 96.8% 115KiB / 9.8%
───────────────────────────────────── ────────────────────── ────────────────────────
Section ncalls time %tot avg alloc %tot avg
───────────────────────────────────────────────────────────────────────────────────────────
parabolic rhs! 1.20k 50.4ms 70.7% 42.0μs 7.19KiB 64.0% 6.13B
├─ calculate gradient 1.20k 26.9ms 37.7% 22.4μs 2.62KiB 23.4% 2.24B
│ ├─ volume integral 1.20k 14.7ms 20.6% 12.2μs ∅ ∅ ∅
│ ├─ Jacobian 1.20k 4.72ms 6.6% 3.93μs ∅ ∅ ∅
│ ├─ surface integral 1.20k 3.35ms 4.7% 2.79μs ∅ ∅ ∅
│ ├─ prolong2interfaces 1.20k 1.21ms 1.7% 1.01μs ∅ ∅ ∅
│ ├─ interface flux 1.20k 1.06ms 1.5% 884ns ∅ ∅ ∅
│ ├─ reset gradients 1.20k 816μs 1.1% 680ns ∅ ∅ ∅
│ ├─ ~calculate gradient~ 1.20k 687μs 1.0% 572ns 2.62KiB 23.4% 2.24B
│ ├─ prolong2boundaries 1.20k 177μs 0.2% 147ns ∅ ∅ ∅
│ ├─ boundary flux 1.20k 111μs 0.2% 92.5ns ∅ ∅ ∅
│ ├─ prolong2mortars 1.20k 38.1μs 0.1% 31.8ns ∅ ∅ ∅
│ └─ mortar flux 1.20k 36.1μs 0.1% 30.1ns ∅ ∅ ∅
├─ volume integral 1.20k 15.7ms 22.0% 13.1μs ∅ ∅ ∅
├─ calculate parabolic fluxes 1.20k 1.71ms 2.4% 1.42μs ∅ ∅ ∅
├─ prolong2interfaces 1.20k 1.31ms 1.8% 1.09μs ∅ ∅ ∅
├─ surface integral 1.20k 1.18ms 1.7% 983ns ∅ ∅ ∅
├─ interface flux 1.20k 1.03ms 1.4% 861ns ∅ ∅ ∅
├─ ~parabolic rhs!~ 1.20k 937μs 1.3% 781ns 4.56KiB 40.6% 3.89B
├─ transform variables 1.20k 749μs 1.1% 624ns ∅ ∅ ∅
├─ boundary flux 1.20k 242μs 0.3% 201ns ∅ ∅ ∅
├─ Jacobian 1.20k 215μs 0.3% 179ns ∅ ∅ ∅
├─ reset ∂u/∂t 1.20k 187μs 0.3% 156ns ∅ ∅ ∅
├─ prolong2boundaries 1.20k 153μs 0.2% 128ns ∅ ∅ ∅
├─ prolong2mortars 1.20k 38.4μs 0.1% 32.0ns ∅ ∅ ∅
├─ mortar flux 1.20k 36.4μs 0.1% 30.4ns ∅ ∅ ∅
└─ source terms parabolic 1.20k 27.3μs 0.0% 22.7ns ∅ ∅ ∅
rhs_hyperbolic! 1.20k 20.9ms 29.3% 17.4μs 4.05KiB 36.0% 3.45B
├─ volume integral 1.20k 15.2ms 21.4% 12.7μs ∅ ∅ ∅
├─ interface flux 1.20k 1.76ms 2.5% 1.47μs ∅ ∅ ∅
├─ surface integral 1.20k 1.18ms 1.6% 980ns ∅ ∅ ∅
├─ prolong2interfaces 1.20k 988μs 1.4% 823ns ∅ ∅ ∅
├─ ~rhs_hyperbolic!~ 1.20k 732μs 1.0% 610ns 4.05KiB 36.0% 3.45B
├─ boundary flux 1.20k 360μs 0.5% 300ns ∅ ∅ ∅
├─ Jacobian 1.20k 217μs 0.3% 181ns ∅ ∅ ∅
├─ reset ∂u/∂t 1.20k 191μs 0.3% 159ns ∅ ∅ ∅
├─ prolong2boundaries 1.20k 136μs 0.2% 113ns ∅ ∅ ∅
├─ mortar flux 1.20k 38.5μs 0.1% 32.1ns ∅ ∅ ∅
├─ prolong2mortars 1.20k 37.1μs 0.1% 30.9ns ∅ ∅ ∅
└─ source terms 1.20k 26.7μs 0.0% 22.2ns ∅ ∅ ∅
───────────────────────────────────────────────────────────────────────────────────────────We can now visualize the solution, which develops a boundary layer at the outflow boundaries.
using Plotsplot(sol)
Package versions
These results were obtained using the following versions.
using InteractiveUtilsversioninfo()using PkgPkg.status(["Trixi", "OrdinaryDiffEqLowStorageRK", "Plots"], mode = PKGMODE_MANIFEST)Julia Version 1.10.12
Commit d93beab124c (2026-08-15 10:29 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 4 × AMD EPYC 9V45 96-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, generic)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
Environment:
JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
Status `~/work/Trixi.jl/Trixi.jl/docs/Manifest.toml`
[b0944070] OrdinaryDiffEqLowStorageRK v3.4.0
[91a5bcdd] Plots v1.41.7
[a7f1ee26] Trixi v0.17.12 `~/work/Trixi.jl/Trixi.jl`This page was generated using Literate.jl.