Abstract

This work forms the foundation for addressing high-order immersed interface methods to solve interface problems and enables us to conduct in-depth examination of this theory. Here, we focus on the introduction a fourth-order finite-difference formulation to approximate the second-order derivative of discontinuous functions. The approach is based on the combination of a high-order implicit formulation and the immersed interface method. The idea is to modify the standard schemes by introducing additional contribution terms based on jump conditions. These contributions are calculated only at grid points where the stencil intersects with the interface. Here, we discuss the issues of implementing the one-dimensional Poisson equation and the heat conduction equation with discontinuous solutions as a three-point stencil for each grid point on the computational domain. In both cases, the resulting discretization approach yields a tridiagonal linear system with matrix coefficients identical to those employed for smooth solutions. We present several numerical experiments to verify the feasibility and accuracy of the method. Thus, this high-order method provides an attractive numerical framework that can efficiently lead to the solution to more complex problems.