On the Regularity of Approximate Solutions to Conservation Laws with Piecewise Smooth Solutions

In this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted W 1,1 space. Convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise-error estimate is given. In other words, we are able to convert weak pointwise-error estimates to optimal error bounds in a weighted W 1,1 space. For convex conservation laws, the assumption of a weak pointwise-error estimate is verified by Tadmor [SIAM J. Numer. Anal., 28 (1991), pp. 891–906]. Therefore, one immediate application of our W 1,1convergence theory is that for convex conservation laws we indeed have W 1,1-error bounds for the approximate solutions to conservation laws. Furthermore, the O( )-pointwise-error estimates of Tadmor and Tang [SIAM J. Numer. Anal., 36 (1999), pp. 1739–1758] are recovered by the use of the W 1,1-convergence result.