Asymptotic estimates of green's function and the “difference step” in the case of Lipschitz-continuous coefficients

Abstract

A study is made of the elementary hyperbolic equation u t ϱ( ζ) u ζ =0 ϱ( ζ) is assumed to be Lipschitz continuous. Asymptotic estimates are obtained for Green's function and the “difference step” for difference schemes of maximum odd order of accuracy 2 k−1, k= O(lnτ−1), τ is the time step. The natural constraint g9( ζ) τ/ h⩽1 is imposed on the ratio of mesh steps. The basic asymptotic estimates are obtained by the saddle-point method. The main difficulty lies in the fact that, with n⪢ k ω , the integrand contains, instead of the power of a single function, as in the case of constant coefficients, the product of essentially different factors. Using the Lipschitz continuity, we obtain estimates in L 1, close to optimal, and identical with the estimates for the case of constant coefficients.