Solving 2D poisson equation in polar coordinates (finite differences)

Aran
Sun, 09 Jan 2022 12:47:58 +0000
Numerics

Hello all,
I am looking to solve the following poisson equation
\nabla^2 \psi = -\omega \quad \text{and} \quad \nabla^2 = \frac{\partial ^2}{{\partial \rho^2 }} + \frac{1}{\rho }\frac{\partial}{\partial \rho} + \frac{1}{{\rho ^2 }}\frac{\partial ^2 }{{\partial \phi ^2 }}.

Where \omega is known over a rectangular domain of grid points of N by M .
I am using finite differences to represent the derivatives of \psi

I am curious whether there are any pre-written Poisson solvers, I have seen a related post below, but didn’t know how to apply any of the recommendations. I have experimented a little with sparse matrices and made a solver on my own, however I would much rather adapt something that is pre-written and that has been verified to be working correctly and efficiently.
Maybe something from ApproxFun.jl or DifferentialEquations.jl?

How to tackle 2D Poisson equation? Numerics

Can anyone point me in the right direction for solving a 2D Poisson equation in a circular region? I’m a little overwhelmed by the number of different Julia packages which a google search returns, and it can be hard to work out what’s current, which packages are abandoned or superseded by others, etc. Ideally I’m looking for something which is Julia all the way through, rather than a wrapper for a third party application. I don’t know much about numerical methods for differential equations, h…

Any help is very much appreciated.
Many thanks

21 posts - 9 participants