A compact sixth-order implicit immersed interface method to solve 2D Poisson equations with discontinuities

Abstract

This paper proposes a compact sixth-order accurate numerical method to solve Poisson equations with discontinuities across an interface. This scheme is based on two techniques for the second-order derivative approximation: a high-order implicit finite difference (HIFD) formula to increase the precision and an immersed interface method (IIM) to deal with the discontinuities. The HIFD formulation arises from Taylor series expansion, and the new formulas are simple modifications to the standard finite difference schemes. On the other hand, the IIM allows one to solve the differential equation using a fixed Cartesian grid by adding some correction terms only at grid points near the immersed interface. The two-dimensional equation is then solved by a nine-point compact sixth-order scheme named HIFD-IIM. Fourth- and second-order methods result in particular cases of the proposed method. Furthermore, the sixth-order method requires similar computational resources to a fourth-order formulation because the resulting matrices in both discretizations are the same. However, higher-order methods require the knowledge of more jump conditions at the interface. From the theoretical derivation of the proposed method, we expect fully six-order accuracy in the maximum norm. This order has been confirmed from our numerical experiments using nontrivial analytical solutions.