Abstract: In this paper, we introduce a new Glimm functional for general systems of hyperbolic conservation laws. This new functional is consistent with the classical Glimm functional for the case when each characteristic field is either genuinely nonlinear or linearly degenerate so that it can be viewed as “optimal” in some sense. With this new functional, the consistency of the Glimm scheme is proved clearly for general systems. Moreover, the convergence rate of the Glimm scheme is shown to be the same as the one obtained in [1] for systems with each characteristic field being genuinely nonlinear or linearly degenerate.

[1] Bressan, A., Marson A., Error bounds for a deterministic version of the Glimm scheme, Arch. Rational Mech. Anal. 142 (1998), no. 2, 155–176. (doi:10.1007/s002050050088; see also preprint 1996-032.)