Existence of the discrete travelling waves for a relaxing scheme Original Research Article

The existence of a discrete travelling wave is proved for the relaxing scheme. The main idea is to change the original scheme such that the resulting scheme is monotonic to which Jenningsʹ result can be applied. The equivalence of the resulting scheme and the original one is shown when View the MathML source. The u-component of the discrete travelling wave thus obtained is a discrete shock for a monotone conservative difference scheme, which approximates the corresponding conservation law.