Simple second-order finite differences for elliptic PDEs with discontinuous coefficients and interfaces

Abstract

In multiphase 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 derivation is relatively simple and mainly involves centered finite difference formulas, with less reliance on the Taylor series expansions of typical immersed interface method derivations. 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.