Nonlinear Stability and Existence of Stationary Discrete Travelling Waves for the Relaxing Schemes

In this paper we show the existence and nonlinear stability of strong stationary discrete travelling waves for the explicit and implicit relaxing schemes introduced by Jin and Xin [3]. For the explicit relaxing scheme, we require that the ratio of the size of the time step and the relaxation rate is bounded above for proving the stability. But for the implicit relaxing scheme, we do not need such a restriction. This is comparable to the numerical results obtained in [3]. The proofs involve a detailed study of error equations for perturbations and an elementary but technical energy method.