  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.
