An immersed interface method for nonlinear convection–diffusion equations with interfaces

Abstract

This paper provides an initial framework for developing high-order numerical methods to solve interface problems for nonlinear elliptic partial differential equations. The proposed formulation is based on the immersed interface method dealing with a discontinuous coefficient problem. The algorithm introduces new schemes for points near the interface, whereas standard central finite difference schemes are used in smooth regions. As a consequence, a global second-order accurate solution is guaranteed. First, theoretical results on the truncation error are given for one-dimensional linear problems. Next, the algorithm is generalized to deal with nonlinear convection and diffusion cases using the using Levenberg–Marquardt algorithm. Numerical simulations for several benchmark problems show the robustness and efficiency of the proposed scheme.