Abstract: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 iderivativeof the approximate solutions is established under the assumption that a weakpointwise-errorestimate 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 $W^{1,1}$- convergence theory is that for convex conservation laws we indeed have $W^{1,1}$-error bounds for the approximate solutions to conservation laws. Furthermore, the $\cO(\en)$-pointwise error estimates of Tadmor and Tang [SIAM J. Numer. Anal., 36 (1999), pp. 1739-1758] is recovered by the use of our $W^{1,1}$-convergence theory.

**Paper:**- Available as PostScript.
**Author(s):**- Tao Tang, <ttang@math.hkbu.edu.hk>
- Z.-H. Teng, <tengzh@sxx0.math.pku.edu.cn>
**Publishing information:****Comments:****Submitted by:**- <ttang@math.hkbu.edu.hk> November 11 1999.

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