Convergence rates to the discrete travelling wave for relaxation schemes

This paper is concerned with the asymptotic convergence of numerical solutions toward discrete travelling waves for a class of relaxation numerical schemes, approximating the scalar conservation law. It is shown that if the initial perturbations possess some algebraic decay in space, then the numerical 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 perturbation of the discrete travelling wave and a technical energy method are applied to obtain the asymptotic convergence rate.