Convergence of a generalized Riemann problem scheme for the Burgers equation

TL;DR

This paper analyzes the convergence of a second order finite volume scheme based on the generalized Riemann problem for the Burgers equation, addressing stability issues and proposing a stabilized method with artificial viscosity.

Contribution

It introduces a stabilized GRP scheme with artificial viscosity and proves its convergence under boundedness assumptions, improving the scheme's stability.

Findings

The original GRP scheme may be entropy unstable with shocks.

Adding artificial viscosity stabilizes the scheme.

The stabilized scheme is proven to be consistent and convergent.

Abstract

In this paper we study the convergence of a second order finite volume approximation of the scalar conservation law. This scheme is based on the generalized Riemann problem (GRP) solver. We firstly investigate the stability of the GRP scheme and find that it might be entropy unstable when the shock wave is generated. By adding an artificial viscosity we propose a new stabilized GRP scheme. Under the assumption that numerical solutions are uniformly bounded, we prove consistency and convergence of this new GRP method.