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 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 Script O sign(ε)-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.
| Original language | English |
|---|---|
| Pages (from-to) | 1483-1495 |
| Number of pages | 13 |
| Journal | SIAM Journal on Numerical Analysis |
| Volume | 38 |
| Issue number | 5 |
| DOIs | |
| Publication status | Published - 2001 |
Scopus Subject Areas
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics
User-Defined Keywords
- Conservation laws
- Error estimates
- Optimal convergence rate
- Viscosity approximation