Simple Second-Order Finite Differences for Elliptic PDEs with Discontinuous Coefficients and Interfaces

TL;DR

This paper introduces a simple, second-order accurate finite difference method for solving elliptic PDEs with discontinuous coefficients and interfaces, suitable for multi-dimensional problems with irregular interfaces.

Contribution

A new finite difference approach that preserves sharp interfaces and handles discontinuities efficiently on Cartesian grids, with proven second-order accuracy in 1D and demonstrated in higher dimensions.

Findings

Achieves second-order accuracy in 1D problems.

Effectively handles discontinuous solutions at interfaces.

Requires only about five iterations to solve the linear system.

Abstract

In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise due to changes in material properties at an immersed interface or embedded boundary, which may have an irregular shape. Consequently, the solution and its gradient can be discontinuous, and numerical methods can be difficult to design. Here a new method is presented and analyzed, using a simple formulation of one-dimensional finite differences on a Cartesian grid, allowing for a relatively easy setup for one-, two-, or three-dimensional problems. The method preserves a sharp interface with discontinuous solutions, obtained from a small number of iterations (approximately five) of solving a symmetric linear system with updates to the right- hand side. Second-order accuracy is rigorously proven in one spatial dimension and demonstrated through numerical examples in two and three spatial dimensions. The method is tested here on the variable-coefficient Poisson equation, and it could be extended for use on time-dependent problems of heat transfer, fluid dynamics, or other applications.