A finite difference Poisson solver for irregular geometries

Abstract

The motivation for this work comes from the development of a 3D quasi-geostrophic Contour Advective Semi-Lagrangian model for vortex interaction in the ocean. The existing code is limited to circular cylindrical geometry and uses polar coordinates. We wish to extend the method to more general cross-sections. The crucial aspect is the solution of the Poisson equation that allows the determination of the stream function from the potential vorticity at each time-step, as this is the part of the algorithm that must be performed on a grid: the advection of potential vorticity contours is fully Lagrangian and hence is easily modified for irregular domains. We develop a 2D algorithm for inverting the Poisson equation for the stream function on an arbitrarily shaped domain, in the special case when the boundary is a streamline, as is the case for our problem. However, the method is also valid for non-zero Dirichlet boundary conditions. The approach uses finite differences with the domain embedded in a rectangular Cartesian grid. We show that the algorithm is second-order accurate and provide several numerical examples.