Implicit Difference Methods for First-Order Partial Differential Functional Equations

Abstract

We present a new class of numerical methods for quasilinear first-order partial differential functional equations. The numerical methods are difference schemes implicit with respect to time variable. We give a complete convergence analysis for the methods and show by an example that the new methods are considerably better than explicit schemes. The proof of stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators with respect to the functional variable.