Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation

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 first 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 the consistency and convergence of this new GRP method.