Introduction

This is the documentation for FiniteVolumeMethod.jl. Click here to go back to the GitHub repository.

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

\[\pdv{u(\vb x, t)}{t} + \div \vb q(\vb x, t, u) = S(\vb x, t, u), \quad (x, y)^{\mkern-1.5mu\mathsf{T}} \in \Omega \subset \mathbb R^2,\,t>0,\]

using the finite volume method, with additional support for steady-state problems and for systems of PDEs of the above form. We support Neumann, Dirichlet, and boundary conditions on $\mathrm du/\mathrm dt$, as well as internal conditions and custom constraints. We also provide an interface for solving special cases of the above PDE, namely reaction-diffusion equations

\[\pdv{u(\vb x, t)}{t} = \div\left[D(\vb x, t, u)\grad u(\vb x, t)\right] + S(\vb x, t, u).\]

The tutorials in the sidebar demonstrate the many possibilities of this package. In addition to these two generic forms, we also provide support for specific problems that can be solved in a more efficient manner, namely:

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

See the Solvers for Specific Problems, and Writing Your Own section for more information on these templates.