An improved convergence rate of Glimm scheme for general systems of hyperbolic conservation laws

We study the convergence rate of Glimm scheme for general systems of hyperbolic conservation laws without the assumption that each characteristic field is either genuinely nonlinear or linearly degenerate. We first give a sharper estimate of the error arising from the wave tracing argument by a careful analysis of the interaction between small waves. With this key estimate, the convergence rate is shown to be , which is sharper compared to given in [T. Yang, Convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, Taiwanese J. Math. 7 (2) (2003) 195–205]. However, it is still slower than given in [A. Bressan, A. Marson, Error bounds for a deterministic version of the Glimm scheme, Arch. Ration. Mech. Anal. 142 (2) (1998) 155–176] for systems with each characteristic field being genuinely nonlinear or linearly degenerate. Here s is the mesh size and α is any positive constant.