TY - JOUR

T1 - On the Regularity of Approximate Solutions to Conservation Laws with Piecewise Smooth Solutions

AU - Tang, Tao

AU - Teng, Zhen Huan

N1 - Funding information:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong (ttang@math.hkbu.edu.hk). The research of this author was supported by Hong Kong Baptist University and the Research Grants Council of Hong Kong.
*Department of Mathematics, Peking University, Beijing 100871, People’s Republic of China (tengzh@sxx0.math.pku.edu.cn). The research of this author was supported by the China State Major Key Project for Basic Research and Hong Kong Baptist University Research grant FRG/98-99/II-14. Part of this work was carried out while this author was visiting HKBU.
Publisher copyright:
Copyright © 2000 Society for Industrial and Applied Mathematics

PY - 2000/11/17

Y1 - 2000/11/17

N2 - In this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted W1,1 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 W1,1 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 W1,1- convergence theory is that for convex conservation laws we indeed have W1,1-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] are recovered by the use of the W1,1-convergence result.

AB - In this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted W1,1 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 W1,1 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 W1,1- convergence theory is that for convex conservation laws we indeed have W1,1-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] are recovered by the use of the W1,1-convergence result.

KW - Conservation laws

KW - Error estimates

KW - Optimal convergence rate

KW - Viscosity approximation

UR - http://www.scopus.com/inward/record.url?scp=0042694326&partnerID=8YFLogxK

U2 - 10.1137/S0036142999364078

DO - 10.1137/S0036142999364078

M3 - Article

AN - SCOPUS:0042694326

SN - 0036-1429

VL - 38

SP - 1483

EP - 1495

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

IS - 5

ER -