Convergence rate of the Glimm scheme

In this paper we prove that there exists a random sequence \THETA_i for the Glimm scheme such that the approximate solution u^\epsilon(t) converges to the exact semigroup solution S_t \bar u of the strictly hyperbolic system of conservation laws u_t + f(u)_x = 0, u(t = 0) = \bar u as follows: for all T ≥ 0 it holds lim \epsilon→0 ∥u_\epsilon (T )− S_T \bar u∥_{L^1} / (√\epsilon| log \epsilon|) = 0. This result is the extension of the analysis of [8] to the general case, when no assumptions on the flux f are made besides strict hyperbolicity. As a corollary, we obtain a deterministic version of the Glimm scheme for the general system case, extending the analysis of [14]. The analysis requires an extension of the quadratic interaction estimates obtained in [3] in order to analyze interaction occurring during an interval of time.