Residual Viscosity Stabilized RBF-FD Methods for Solving Nonlinear Conservation Laws

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.

Automated summary

This paper presents a radial basis function generated finite difference method for solving nonlinear conservation laws, addressing the issue of numerical instabilities through the introduction of a residual-based artificial viscosity stabilization framework. Computational tests confirm the reliability and accuracy of the proposed stabilized methods in solving both scalar conservation laws and conservation law systems.