High Order Weighted Extrapolation for Boundary Conditions for Finite Difference Methods on Complex Domains with Cartesian Meshes

TL;DR

This paper introduces a new weighted extrapolation method for boundary conditions in finite difference methods on complex domains, improving robustness and stability especially near discontinuities.

Contribution

It presents a novel weighted extrapolation technique combined with least squares stabilization, enhancing boundary condition handling on Cartesian meshes.

Findings

Improved robustness in handling discontinuities.

Enhanced stability of the extrapolation scheme.

Better performance compared to previous methods.

Abstract

The design of numerical boundary conditions is a challenging problem that has been tackled in different ways depending on the nature of the problem and the numerical scheme used to solve it. In this paper we present a new weighted extrapolation technique which entails an improvement with respect to the technique that was developed in [1]. This technique is based on the application of a variant of the Lagrange extrapolation through the computation of weights capable of detecting regions with discontinuities. We also present a combination of the above technique with a least squares approach in order to stabilize the scheme in some cases where Lagrange extrapolation can turn the scheme mildly unstable. We show that this combined extrapolation technique can tackle discontinuities more robustly than the procedure introduced in [1].