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.

翻译：本文研究标量守恒律的二阶有限体积逼近格式的收敛性。该格式基于广义Riemann问题（GRP）求解器。我们首先考察GRP格式的稳定性，发现激波生成时该格式可能因熵不稳定性而失效。通过引入人工黏性，提出一种新的稳定化GRP格式。在数值解一致有界的假设下，我们证明了该新GRP格式的相容性与收敛性。