Generalized Euler method for Hamilton Jacobi differential functional systems

Nonlinear first order partial functional differential systems are considered in the paper. Classical solutions of the local Cauchy problem on the Haar pyramid are approximated by solutions of suitable quasilinear systems of difference functional equations. The proposed numerical methods are difference schemes of the Euler type. A complete convergence analysis is given and it is shown by examples that the new methods are considerable better than the classical methods. It is shown that the Lax scheme is superfluous for the numerical approximations of classical solutions to nonlinear functional differential problems. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators.