First-order L1-convergence for relaxation approximations to conservation laws

✍ Scribed by Zhen–Huan Teng

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 518 KB
• Volume: 51
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/(sici)1097-0312(199808)51:8<857::aid-cpa1>3.0.co;2-4

✦ Synopsis

We derive a first-order rate of L 1 -convergence for stiff relaxation approximations to its equilibrium solutions, i.e., piecewise smooth entropy solutions with finitely many discontinuities for scalar, convex conservation laws. The piecewise smooth solutions include initial central rarefaction waves, initial shocks, possible spontaneous formation of shocks in a future time, and interactions of all these patterns. A rigorous analysis shows that the relaxation approximations to approach the piecewise smooth entropy solutions have L 1 -error bound of O(ε| log ε| + ε), where ε is the stiff relaxation coefficient. The first-order L 1 -convergence rate is an improvement on the O( √ ε) error bound. If neither central rarefaction waves nor spontaneous shocks occur, the error bound is improved to O(ε).