High order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries

TL;DR

This paper introduces a fourth order ghost point finite difference method for solving the Poisson equation on Cartesian grids with curved boundaries, extending previous second order methods and analyzing their stability and accuracy.

Contribution

The paper presents a high order (fourth order) ghost point finite difference scheme for elliptic problems, including stability analysis and numerical verification, extending prior second order methods.

Findings

Two of three discretizations are stable

Stable methods achieve high accuracy in tests

Numerical results confirm theoretical accuracy

Abstract

In this paper a fourth order finite difference ghost point method for the Poisson equation on regular Cartesian mesh is presented. The method can be considered the high order extension of the second ghost method introduced earlier by the authors. Three different discretizations are considered, which differ in the stencil that discretizes the Laplacian and the source term. It is shown that only two of them provide a stable method. The accuracy of such stable methods are numerically verified on several test problems.