FiniteVolumeMethod
Solver for two-dimensional conservation equations using the finite volume method in Julia.
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Solver for two-dimensional conservation equations using the finite volume method in Julia.
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- Stars: 46
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- Open Issues: 1
- Releases: 26
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README.md
FiniteVolumeMethod
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:
DiffusionEquations: $\partial_tu = \boldsymbol\nabla\boldsymbol\cdot[D(\boldsymbol x)\boldsymbol\nabla u]$.MeanExitTimeProblems: $\boldsymbol\nabla\boldsymbol\cdot[D(\boldsymbol x)\boldsymbol\nabla T(\boldsymbol x)] = -1$.LinearReactionDiffusionEquations: $\partial_tu = \boldsymbol\nabla\boldsymbol\cdot[D(\boldsymbol x)\boldsymbol\nabla u] + f(\boldsymbol x)u$.PoissonsEquation: $\boldsymbol\nabla\boldsymbol\cdot[D(\boldsymbol x)\boldsymbol\nabla u] = f(\boldsymbol x)$.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.
julia
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
We could have equivalently used the DiffusionEquation template, so that prob could have also been defined by
julia
prob = DiffusionEquation(mesh, BCs; diffusion_function = D, initial_condition, final_time)
and be solved much more efficiently. See the documentation for more information.
Owner
- Name: SciML Open Source Scientific Machine Learning
- Login: SciML
- Kind: organization
- Email: contact@chrisrackauckas.com
- Website: https://sciml.ai
- Twitter: SciML_Org
- Repositories: 170
- Profile: https://github.com/SciML
Open source software for scientific machine learning
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Last Year
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Committers
Last synced: 11 months ago
Top Committers
| Name | Commits | |
|---|---|---|
| DanielVandH | d****l@g****m | 187 |
| github-actions[bot] | 4****] | 1 |
| Leandro Martínez | 3****q | 1 |
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Last synced: 9 months ago
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Past Year
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- Average time to close issues: N/A
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Top Authors
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- JuliaTagBot (1)
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- DanielVandH (7)
- ChrisRackauckas (1)
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Packages
- Total packages: 1
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Total downloads:
- julia 2 total
- Total dependent packages: 0
- Total dependent repositories: 0
- Total versions: 31
juliahub.com: FiniteVolumeMethod
Solver for two-dimensional conservation equations using the finite volume method in Julia.
- Documentation: https://docs.juliahub.com/General/FiniteVolumeMethod/stable/
- License: MIT
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Latest release: 1.1.5
published over 1 year ago
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- JuliaRegistries/TagBot v1 composite
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