Preprint 1999-036

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

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 i 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 $W^{1,1}$- 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>

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<ttang@math.hkbu.edu.hk> November 11 1999.