A fourth-order compact implicit immersed interface method for 2D Poisson interface problems

Abstract

This paper presents a fourth-order compact immersed interface method to solve two-dimensional Poisson equations with discontinuous solutions on arbitrary domains divided by an interface. The compact scheme only employs a nine-point stencil for each grid point on the computational domain. The new approach is based on an implicit formulation obtained from generalized Taylor series expansions, and it is constructed from a few modifications to the central finite difference near the interface. The discretization results in a linear system in which the matrix coefficients are the same as the ones for smooth solutions, and the right-hand side system is modified by adding terms known as jump contributions. These contributions are only calculated at those points where the nine-point stencil cuts the interface. However, the contribution formulas require the knowledge of Cartesian jumps up to fourth-order. In this paper, we derived them using only the principal jump conditions and the jumps coming from the known right-hand function of the Poisson equation. We present numerical experiments in two dimensions to verify the feasibility and accuracy of the proposed method. Thus, the implicit immersed interface method results in an attractive fourth-order compact scheme that is easy to be implemented and applied to arbitrary interface shapes.