10: Adding a new scalar conservation law

If you want to use Trixi.jl for your own research, you might be interested in a new physics model that's not already included in Trixi.jl. In this tutorial, we will implement the cubic conservation law

\[\partial_t u(t,x) + \partial_x u(t,x)^3 = 0\]

in a periodic domain in one space dimension. In Trixi.jl, such a mathematical model is encoded as a subtype of Trixi.AbstractEquations.

Basic setup

using Trixi

struct CubicEquation <: Trixi.AbstractEquations{1 #= number of spatial dimensions =#,
                                                1 #= number of primary variables, i.e. scalar =#};
end

We create CubicEquation as an empty struct since we do not use any parameters for this equation. Other models could bundle arbitrary parameters, e.g., the ideal gas constant for the compressible Euler equations.

Next, we define the physical flux f(u) = u^3 using the calling structure used in Trixi.jl.

Trixi.flux(u, orientation, equation::CubicEquation) = u.^3
Trixi.varnames(_, ::CubicEquation) = ("scalar",)

In Trixi.jl, the conserved variables u are usually passed as SVectors of variables at a single physical location. Hence, we must use u.^3 instead of the scalar operation u^3.

That's already enough to run a simple simulation with a standard DGSEM discretization using the non-dissipative central flux at interfaces.

using OrdinaryDiffEq

# Create a simulation setup
equation = CubicEquation()

initial_condition_sine(x, t, equation::CubicEquation) = SVector(sinpi(x[1]))

mesh = TreeMesh(-1.0, 1.0, # min/max coordinates
                initial_refinement_level=4,
                n_cells_max=10^4)

solver = DGSEM(3 #= polynomial degree =#, flux_central)

semi = SemidiscretizationHyperbolic(mesh, equation, initial_condition_sine, solver)

┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ SemidiscretizationHyperbolic                                                                     │
│ ════════════════════════════                                                                     │
│ #spatial dimensions: ………………………… 1                                                                │
│ mesh: ………………………………………………………………… TreeMesh{1, Trixi.SerialTree{1}} with length 31                  │
│ equations: …………………………………………………… CubicEquation                                                    │
│ initial condition: ……………………………… initial_condition_sine                                           │
│ boundary conditions: ………………………… Trixi.BoundaryConditionPeriodic                                  │
│ source terms: …………………………………………… nothing                                                          │
│ solver: …………………………………………………………… DG                                                               │
│ total #DOFs per field: …………………… 64                                                               │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘

We wrap the return value of the initial_condition_sine inside an SVector since that's the approach used in Trixi.jl also for systems of equations. We need to index the spatial coordinate x[1], since it is an SVector with one component. In multiple space dimensions, all spatial coordinates are passed together.

Next, we create an ODEProblem from the SciML/DifferentialEquations ecosystem. We can solve this ODE numerically using any time integration method, e.g., SSPRK43 from OrdinaryDiffEq.jl. Before, we set up a callback to summarize the simulation setup.

# Create ODE problem with given time span
tspan = (0.0, 0.09)
ode = semidiscretize(semi, tspan)

summary_callback = SummaryCallback()
callbacks = CallbackSet(summary_callback)

# OrdinaryDiffEq's `solve` method evolves the solution in time and executes the passed callbacks
sol = solve(ode, SSPRK43();
            ode_default_options()..., callback=callbacks);


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   ██║   ██████╔╝██║ ╚███╔╝ ██║
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   ╚═╝   ╚═╝  ╚═╝╚═╝╚═╝  ╚═╝╚═╝

┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ SemidiscretizationHyperbolic                                                                     │
│ ════════════════════════════                                                                     │
│ #spatial dimensions: ………………………… 1                                                                │
│ mesh: ………………………………………………………………… TreeMesh{1, Trixi.SerialTree{1}} with length 31                  │
│ equations: …………………………………………………… CubicEquation                                                    │
│ initial condition: ……………………………… initial_condition_sine                                           │
│ boundary conditions: ………………………… Trixi.BoundaryConditionPeriodic                                  │
│ source terms: …………………………………………… nothing                                                          │
│ solver: …………………………………………………………… DG                                                               │
│ total #DOFs per field: …………………… 64                                                               │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘

┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ TreeMesh{1, Trixi.SerialTree{1}}                                                                 │
│ ════════════════════════════════                                                                 │
│ center: …………………………………………………………… [0.0]                                                            │
│ length: …………………………………………………………… 2.0                                                              │
│ periodicity: ……………………………………………… (true,)                                                          │
│ current #cells: ……………………………………… 31                                                               │
│ #leaf-cells: ……………………………………………… 16                                                               │
│ maximum #cells: ……………………………………… 10000                                                            │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘

┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ CubicEquation                                                                                    │
│ ═════════════                                                                                    │
│ #variables: ………………………………………………… 1                                                                │
│ │ variable 1: …………………………………………… scalar                                                           │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘

┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ DG{Float64}                                                                                      │
│ ═══════════                                                                                      │
│ basis: ……………………………………………………………… LobattoLegendreBasis{Float64}(polydeg=3)                         │
│ mortar: …………………………………………………………… LobattoLegendreMortarL2{Float64}(polydeg=3)                      │
│ surface integral: ………………………………… SurfaceIntegralWeakForm                                          │
│ │ surface flux: ……………………………………… flux_central                                                     │
│ volume integral: …………………………………… VolumeIntegralWeakForm                                           │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘

┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ Time integration                                                                                 │
│ ════════════════                                                                                 │
│ Start time: ………………………………………………… 0.0                                                              │
│ Final time: ………………………………………………… 0.09                                                             │
│ time integrator: …………………………………… SSPRK43                                                          │
│ adaptive: ……………………………………………………… true                                                             │
│ abstol: …………………………………………………………… 1.0e-6                                                           │
│ reltol: …………………………………………………………… 0.001                                                            │
│ controller: ………………………………………………… PIController{Rational{Int64}}(7//30, 2//15)                      │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘
┌──────────────────────────────────────────────────────────────────────────────────────────────────┐
│ Environment information                                                                          │
│ ═══════════════════════                                                                          │
│ #threads: ……………………………………………………… 1                                                                │
└──────────────────────────────────────────────────────────────────────────────────────────────────┘

That's it, you ran your first simulation using your new equation with Trixi.jl! Now, we can plot the solution at the final time using Plots.jl.

using Plots
plot(sol)

You can already see that discontinuities will develop and oscillations start to occur around steep parts of the wave. That's expected from our central discretization. To avoid these issues, we need to use dissipative numerical fluxes (approximate Riemann solvers) at interfaces.

Advanced setup

Thus, we add a Godunov's flux for our cubic equation. That is easy for this equation since the wave speed f'(u) = 3u^2 is always non-negative.

@inline Trixi.flux_godunov(u_ll, u_rr, orientation, equation::CubicEquation) = flux(u_ll, orientation, equation)

Let's run the example again but with a dissipative numerical flux at interfaces. remake will recreate the semidiscretization we used before and only change selected parameters, in this case the solver.

# A new setup with dissipation
semi = remake(semi, solver=DGSEM(3, flux_godunov))
ode = semidiscretize(semi, tspan)
sol = solve(ode, SSPRK43(); ode_default_options()...)
plot!(sol)

You can see that there are fewer oscillations, in particular around steep edges. Now let's increase the final time (and also the spatial resolution).

# A larger final time: Nonclassical shocks develop (you can even increase the refinement to 12)
semi = remake(semi, mesh=TreeMesh(-1.0, 1.0, initial_refinement_level=8, n_cells_max=10^5))
ode = semidiscretize(semi, (0.0, 0.5) #= tspan =#)
sol = solve(ode, SSPRK43(); ode_default_options()...)
plot(sol)

You can observe that nonclassical shocks develop and are stable under grid refinement, e.g. for initial_refinement_level=12. In this case, these nonclassical shocks can be avoided by using an entropy-dissipative semidiscretization. Thus, we need to define an entropy-conservative numerical flux

@inline function Trixi.flux_ec(u_ll, u_rr, orientation, equation::CubicEquation)
  return SVector(0.25 * (u_ll[1]^3 + u_ll[1]^2 * u_rr[1] + u_ll[1] * u_rr[1]^2 + u_rr[1]^3))
end

and use a VolumeIntegralFluxDifferencing instead of the standard VolumeIntegralWeakForm in the DGSEM.

# Let's use a provably entropy-dissipative semidiscretization
semi = remake(semi, solver=DGSEM(3, flux_godunov, VolumeIntegralFluxDifferencing(flux_ec)))
ode = semidiscretize(semi, (0.0, 0.5))
sol = solve(ode, SSPRK43(); ode_default_options()...);
plot(sol)

Possible next steps could be

to define Trixi.max_abs_speeds(u, equations::CubicEquation) = 3 * u[1]^2 to use CFL-based time step control via a StepsizeCallback
to define quantities of interest like Trixi.entropy(u, equations::CubicEquation) = u[1]^2 and integrate them in a simulation using the AnalysisCallback
to experiment with shock-capturing volume integrals VolumeIntegralShockCapturingHG and adaptive mesh refinement AMRCallback

For further reading, Trixi.jl provides another example on adding a scalar equation. In the elixir about the KPP problem, the 2D scalar "KPP equation" from Kurganov, Petrova, Popov (2007) is implemented.

Summary of the code

To sum up, here is the complete code that we used (without the callbacks since these create a lot of unnecessary output in the doctests of this tutorial). In addition, we create the struct inside the new module CubicConservationLaw. That ensures that we can re-create structs defined therein without having to restart Julia.

# Define new physics
module CubicConservationLaw

using Trixi

struct CubicEquation <: Trixi.AbstractEquations{1 #= number of spatial dimensions =#,
                                                1 #= number of primary variables, i.e. scalar =#}
end

@inline Trixi.flux(u, orientation, equation::CubicEquation) = u.^3
Trixi.varnames(_, ::CubicEquation) = ("scalar",)

@inline Trixi.flux_godunov(u_ll, u_rr, orientation, equation::CubicEquation) = flux(u_ll, orientation, equation)
@inline function Trixi.flux_ec(u_ll, u_rr, orientation, equation::CubicEquation)
  return SVector(0.25 * (u_ll[1]^3 + u_ll[1]^2 * u_rr[1] + u_ll[1] * u_rr[1]^2 + u_rr[1]^3))
end

end # module


# Create a simulation setup
import .CubicConservationLaw
using Trixi
using OrdinaryDiffEq
using Plots

equation = CubicConservationLaw.CubicEquation()

initial_condition_sine(x, t, equation::CubicConservationLaw.CubicEquation) = SVector(sinpi(x[1]))

mesh = TreeMesh(-1.0, 1.0, # min/max coordinates
                initial_refinement_level=4,
                n_cells_max=10^4)

solver = DGSEM(3 #= polynomial degree =#, flux_central)

semi = SemidiscretizationHyperbolic(mesh, equation, initial_condition_sine, solver)

# Create ODE problem with given time span
tspan = (0.0, 0.1)
ode = semidiscretize(semi, tspan)

# OrdinaryDiffEq's `solve` method evolves the solution in time and executes the passed callbacks
sol = solve(ode, SSPRK43(); ode_default_options()...)
plot(sol)


# A new setup with dissipation
semi = remake(semi, solver=DGSEM(3, flux_godunov))
ode = semidiscretize(semi, tspan)
sol = solve(ode, SSPRK43(); ode_default_options()...)
plot!(sol)


# A larger final time: Nonclassical shocks develop (you can even increase the refinement to 12)
semi = remake(semi, mesh=TreeMesh(-1.0, 1.0, initial_refinement_level=8, n_cells_max=10^5))
ode = semidiscretize(semi, (0.0, 0.5))
sol = solve(ode, SSPRK43(); ode_default_options()...)
plot(sol)


# Let's use a provably entropy-dissipative semidiscretization
semi = remake(semi, solver=DGSEM(3, flux_godunov, VolumeIntegralFluxDifferencing(flux_ec)))
ode = semidiscretize(semi, (0.0, 0.5))
sol = solve(ode, SSPRK43(); ode_default_options()...)
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`

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