MethodOfLines.jl: Automated Finite Difference for Physics-Informed Learning

MethodOfLines.jl is a Julia package for automated finite difference discretization of symbolicaly-defined PDEs in N dimensions.

It uses symbolic expressions for systems of partial differential equations as defined with ModelingToolkit.jl, and Interval from DomainSets.jl to define the space(time) over which the simulation runs.

It is a SciML “Discretizer” package, a class of packages which export the methods:

  • discretize(sys::PDESystem, disc::D) where {D <: AbstractDiscretization}, which returns an AbstractSciMLProblem to be solved with the ecosystem's solvers.
  • symbolic_discretize(sys::PDESystem, disc::D) where {D <: AbstractDiscretization}, which returns an AbstractSystem from ModelingToolkit.jl.

A Discretizer also optionally provides automatic solution wrapping, for easing the retrieval of shaped portions of the solution, and multidimensional interpolations. This feature is provided by MethodOfLines.jl, see the solution interface page for more information.

The AbstractDiscretization that MethodOfLines.jl provides is the MOLFiniteDifference, see its documentation for full information about interface options.

The package's handling is quite general, it is recommended to try out your system of equations and post an issue if you run in to trouble. If you want to solve it, we want to support it.

Issues with questions on usage are also welcome as they help us improve the docs.

See here for a full tutorial, involving the Brusselator equation.

Allowable terms in the system include, but are not limited to

  • Advection
  • Diffusion
  • Reaction
  • Nonlinear Diffusion
  • Spherical Laplacian
  • Any Julia function of the symbolic parameters/dependent variables and other parameters in the environment that's defined on the whole domain. Note that more complicated functions may require registration with @register, see the ModelingToolkit.jl docs.

Boundary conditions include, but are not limited to:

  • Dirichlet
  • Neumann (can also include time derivative)
  • Robin (can also include time derivative)
  • Periodic
  • Any Julia function that returns a number

Currently, the centered difference, upwind difference, nonlinear Laplacian and spherical Laplacian schemes are implemented. If you know of a scheme with better stability or accuracy in any specific case, please post an issue with a link to a paper.

Due to an implementation detail, the maximum derivative order that can be discretized by MOL is div(typemax(Int), 2), in 64 bit 4611686018427387903. We hope that this is enough for your purposes!

Installation

Assuming that you already have Julia correctly installed, it suffices to import MethodOfLines.jl in the standard way:

using Pkg
Pkg.add("MethodOfLines")

Contributing

Known Limitations

Currently, the package can discretize almost any system, with some assumptions listed below

  • That the grid is Cartesian.
  • Boundary conditions in time are supplied as initial conditions, not at the end of the simulation interval. If your system requires a final condition, please use a change of variables to rectify this. This is unlikely to change due to upstream constraints.
  • Integral equations are not supported.
  • That dependant variables always have the same argument signature, except in BCs.
  • That higher order interface bcs are accompanied by a simple interface of the form u1(t, x_int) ~ u2(t, x_int)
  • That boundary conditions do not contain references to derivatives which are not in the direction of the boundary, except in time.
  • That odd order derivatives do not multiply or divide each other, unless the WENO Scheme is used.
  • That the WENO scheme must be used when there are mixed derivatives.
  • Note that the WENO Scheme is often unstable in more than 1 spatial dimension due to difficulties with boundary handling, this can be avoided if you supply 2 or more bcs per boundary in the dimension along which an advection term acts.

Reproducibility

The documentation of this SciML package was built using these direct dependencies,
Status `~/work/MethodOfLines.jl/MethodOfLines.jl/docs/Project.toml`
  [0c46a032] DifferentialEquations v7.11.0
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Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`
and using this machine and Julia version.
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
A more complete overview of all dependencies and their versions is also provided.
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  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.0.5+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+0
  [14a3606d] MozillaCACerts_jll v2022.10.11
  [4536629a] OpenBLAS_jll v0.3.21+4
  [05823500] OpenLibm_jll v0.8.1+0
  [efcefdf7] PCRE2_jll v10.42.0+0
  [bea87d4a] SuiteSparse_jll v5.10.1+6
  [83775a58] Zlib_jll v1.2.13+0
  [8e850b90] libblastrampoline_jll v5.8.0+0
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`

You can also download the manifest file and the project file.