A Novel HOC-Immersed Interface Approach For Elliptic Problems

TL;DR

This paper introduces a higher-order accurate finite difference immersed interface method for elliptic problems with complex interfaces, demonstrating improved accuracy and convergence over existing methods through various numerical experiments.

Contribution

A new higher-order immersed interface method with a novel discretization strategy that maintains compactness and improves accuracy for elliptic problems with complex interfaces.

Findings

Achieves higher-order accuracy and convergence rates.

Demonstrates superiority over existing IIM methods.

Results closely match analytical, numerical, and experimental data.

Abstract

We present a new higher-order accurate finite difference explicit jump Immersed Interface Method (HEJIIM) 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 EJIIM and other existing IIM methods by establishing the rate of convergence and grid independence of the computed solutions. In all the cases our computed results extremely close to the available numerical and experimental results.