A novel higher order compact-immersed interface approach for elliptic problems

Abstract

We present a new higher-order accurate finite difference explicit jump Immersed Interface Method for solving two-dimensional elliptic problems with singular source and discontinuous coefficients in the irregular region on a compact Cartesian mesh. We propose a new strategy for discretizing the solution at irregular points on a nine-point compact stencil such that the higher-order compactness is maintained throughout the whole computational domain. The scheme is employed to solve four problems embedded with circular- and star-shaped interfaces in a rectangular region having analytical solutions and varied discontinuities across the interface in source and the coefficient terms. We also simulate a plethora of fluid flow problems past bluff bodies in complex flow situations which are governed by the Navier–Stokes equations; they include problems involving multiple bodies immersed in the flow as well. In the process, we show the superiority of the proposed strategy over the explicit jump finite difference immersed interface method and other existing immersed interface methods by establishing the rate of convergence and grid independence of the computed solutions. In all the cases, our computed results are extremely close to the available numerical and experimental results.