FiniteVolumeMethod.jl

Solver for two-dimensional conservation equations using the finite volume method in Julia.
Author SciML
Popularity
32 Stars
Updated Last
10 Months Ago
Started In
November 2022

FiniteVolumeMethod

DOI Dev Stable Coverage

This is a Julia package for solving partial differential equations (PDEs) of the form

$$ \dfrac{\partial u(\boldsymbol x, t)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{q}(\boldsymbol x, t, u) = S(\boldsymbol x, t, u), \quad (x, y)^{\mkern-1.5mu\mathsf{T}} \in \Omega \subset \mathbb R^2,t>0, $$

in two dimensions using the finite volume method, with support also provided for steady-state problems and for systems of PDEs of the above form. In addition to this generic form above, we also provide support for specific problems that can be solved in a more efficient manner, namely:

  1. DiffusionEquations: $\partial_tu = \boldsymbol\nabla\boldsymbol\cdot[D(\boldsymbol x)\boldsymbol\nabla u]$.
  2. MeanExitTimeProblems: $\boldsymbol\nabla\boldsymbol\cdot[D(\boldsymbol x)\boldsymbol\nabla T(\boldsymbol x)] = -1$.
  3. LinearReactionDiffusionEquations: $\partial_tu = \boldsymbol\nabla\boldsymbol\cdot[D(\boldsymbol x)\boldsymbol\nabla u] + f(\boldsymbol x)u$.
  4. PoissonsEquation: $\boldsymbol\nabla\boldsymbol\cdot[D(\boldsymbol x)\boldsymbol\nabla u] = f(\boldsymbol x)$.
  5. LaplacesEquation: $\boldsymbol\nabla\boldsymbol\cdot[D(\boldsymbol x)\boldsymbol\nabla u] = 0$.

See the documentation for more information.

If this package doesn't suit what you need, you may like to review some of the other PDE packages shown here.

As a very quick demonstration, here is how we could solve a diffusion equation with Dirichlet boundary conditions on a square domain using the standard FVMProblem formulation; please see the docs for more information.

using FiniteVolumeMethod, DelaunayTriangulation, CairoMakie, OrdinaryDiffEq
a, b, c, d = 0.0, 2.0, 0.0, 2.0
nx, ny = 50, 50
tri = triangulate_rectangle(a, b, c, d, nx, ny, single_boundary=true)
mesh = FVMGeometry(tri)
bc = (x, y, t, u, p) -> zero(u)
BCs = BoundaryConditions(mesh, bc, Dirichlet)
f = (x, y) -> y  1.0 ? 50.0 : 0.0
initial_condition = [f(x, y) for (x, y) in DelaunayTriangulation.each_point(tri)]
D = (x, y, t, u, p) -> 1 / 9
final_time = 0.5
prob = FVMProblem(mesh, BCs; diffusion_function=D, initial_condition, final_time)
sol = solve(prob, Tsit5(), saveat=0.001)
u = Observable(sol.u[1])
fig, ax, sc = tricontourf(tri, u, levels=0:5:50, colormap=:matter)
tightlimits!(ax)
record(fig, "anim.gif", eachindex(sol)) do i
    u[] = sol.u[i]
end

Animation of a solution

We could have equivalently used the DiffusionEquation template, so that prob could have also been defined by

prob = DiffusionEquation(mesh, BCs; diffusion_function=D, initial_condition, final_time)

and be solved much more efficiently. See the documentation for more information.