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This paper is concerned with the asymptotic convergence of nu- merical solutions toward discrete travelling waves for a class of relaxation nu- merical schemes, approximating the scalar conservation law. It is shown that if the initial perturbations possess some algebraic decay in space, then the nu- merical solutions converge to the discrete travelling wave at a corresponding algebraic rate in time, provided the sums of the initial perturbations for the u-component equal zero. A polynomially weighted l 2 norm on the perturba- tion of the discrete travelling wave and a technical energy method are applied to obtain the asymptotic convergence rate.

Convergence rates to the discrete travelling wave for relaxation schemes