Residual viscosity stabilized RBF-FD methods for solving nonlinear conservation laws

TL;DR

This paper introduces a residual-based artificial viscosity stabilization for RBF-FD methods, improving their stability and accuracy in solving nonlinear conservation laws with shocks and discontinuities.

Contribution

It develops an adaptive residual-based viscosity framework tailored for RBF-FD methods to handle shocks in nonlinear conservation laws.

Findings

Enhanced stability and accuracy in solving conservation laws.

Effective shock detection and stabilization demonstrated.

Reliable performance on Euler equations.

Abstract

In this paper, we solve nonlinear conservation laws using the radial basis function generated finite difference (RBF-FD) method. Nonlinear conservation laws have solutions that entail strong discontinuities and shocks, which give rise to numerical instabilities when the solution is approximated by a numerical method. We introduce a residual-based artificial viscosity (RV) stabilization framework adjusted to the RBF-FD method, where the residual of the conservation law adaptively locates discontinuities and shocks. The RV stabilization framework is applied to the collocation RBF-FD method and the oversampled RBF-FD method. Computational tests confirm that the stabilized methods are reliable and accurate in solving scalar conservation laws and conservation law systems such as compressible Euler equations.