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Convergence Rates for Relaxation Schemes Approximating Conservation Laws

by Hailiang Liu and Gerald Warnecke.

Series: 1998-31, Preprints

35L65 Conservation laws
65M06 Finite difference methods

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.

relaxation scheme, relaxation model, convergence rate

This paper was published in:
SIAM J. Numer. Anal. 37, No. 4, S. 1316 - 1337, 2000