Sixth Order Compact Finite Difference Scheme for Poisson Interface Problem with Singular Sources

TL;DR

This paper introduces a sixth order compact finite difference scheme for solving Poisson interface problems with singular sources, achieving high accuracy without coordinate transformations or complex stencil formulas.

Contribution

The paper extends the immersed interface method to sixth order accuracy on uniform Cartesian grids with explicit stencils, independent of source terms and interface geometry.

Findings

Numerical experiments confirm sixth order accuracy.

The scheme handles singular sources and discontinuities effectively.

Coefficient matrix is independent of problem specifics.

Abstract

Let $\Gamma$ be a smooth curve inside a two-dimensional rectangular region $\Omega$. In this paper, we consider the Poisson interface problem $-\nabla^2 u=f$ in $\Omega\setminus \Gamma$ with Dirichlet boundary condition such that $f$ is smooth in $\Omega\setminus \Gamma$ and the jump functions $[u]$ and $[\nabla u\cdot \vec{n}]$ across $\Gamma$ are smooth along $\Gamma$. This Poisson interface problem includes the weak solution of $-\nabla^2 u=f+g\delta_\Gamma$ in $\Omega$ as a special case. Because the source term $f$ is possibly discontinuous across the interface curve $\Gamma$ and contains a delta function singularity along the curve $\Gamma$, both the solution $u$ of the Poisson interface problem and its flux $\nabla u\cdot \vec{n}$ are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils. The coefficient matrix $A$ in the resulting linear system $Ax=b$, following from the proposed scheme, is independent of any source term $f$, jump condition $g\delta_\Gamma$, interface curve $\Gamma$ and Dirichlet boundary conditions. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources.