A higher-order finite-volume discretization method for Poisson's equation in cut cell geometries

Abstract

We present a method for generating higher-order finite volume discretizations for Poisson's equation on Cartesian cut cell grids in two and three dimensions. The discretization is in flux-divergence form, and stencils for the flux are computed by solving small weighted least-squares linear systems. Weights are the key in generating a stable discretization. We apply the method to solve Poisson's equation on a variety of geometries, and we demonstrate that the method can achieve second and fourth order accuracy in both truncation and solution error for these examples. We also show that the Laplacian operator has only stable eigenvalues for each of these examples.