A Second‐Order Immersed Interface Method for Degenerate Parabolic Interface Problems

In this paper we consider a second‐order immersed interface method for reaction‐convection‐diffusion partial differential equations with a degenerate diffusion term. We assume that the diffusion coefficient is discontinuous across an interface and vanishes on one of its sides. The proposed formulation consider the degeneration as a discontinuous coefficient interface problem. The numerical solution is computed using standard centered finite difference schemes away from the interface along with a novel scheme specifically designed for the interface region. This new scheme introduces modified finite‐difference coefficients and contribution terms that explicitly incorporate known jump values at the interface. As a consequence, this approach ensures a globally second‐order accurate numerical solution. Theoretical results on truncation error for one‐dimensional degenerate elliptic problems are first presented followed by an analysis of the modifications necessary for achieving accurate numerical solutions in degenerate parabolic problems. Next, the numerical implementation of the two‐dimensional problem is introduced. Finally, numerical examples for several benchmark problems show the robustness, accuracy, and efficiency of the proposed scheme.