We establish here the convergence (thereby proving the existence) of a semi-discrete scheme for the quasilinear hyperbolic equation where x ∈ Rn, t ∈ [0,T], and ? ∈ L∞ (Rn). It is well known that the above problem does not necessarily have global classical solutions and the usual concepts of weak solution. do not lead to a unique solution The existence of a unique solution to the above problem in a suitable sense was established in [3], where a parabolic problem obtained by introducing the term ??Δu was studied and then the behavior as ? → 0 was discussed. A difference scheme approach to a problem of the above type where ?i does not depend on x and t and Ψ does not depend on u was also studied in [2]. The aim of this paper is to present a proof for the case when ? depends on x, Ψ depends on u, and the technical complications in this case are nontrivial. The discussions in this paper my be considered as continuation of the ideas in the above papers.
