Convergence Rates for Relaxation Schemes Approximating Conservation Laws

by Hailiang Liu and Gerald Warnecke


Preprint series: 98-31, Preprints

The paper is published: SIAM J. Numer. Anal. 37, No. 4, S. 1316 - 1337, 2000

35L65 Conservation laws
65M06 Finite difference methods


Abstract: In this paper, we prove a global error estimate for a relaxation scheme approximating scalar conservation laws. To this end, we decompose the error into a relaxation error and a discretization error. Including an initial error $\omega(\epsilon)$ we obtain the rate of convergence of $(\max \{\ep, \omega(\epsilon)\})^{1/2}$ in $L^1$ for the relaxation step. The estimate here is independent of the type of nonlinearity. In the discretization step a convergence rate of $(\Delta x)^{1/2}$ in $L^1$ is obtained and is independent of the choice of initial error $\omega(\ep)$. Thereby, we obtain the order $1/2$ for the total error.

Keywords: relaxation scheme, relaxation model, convergence rate

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