Generalized Euler Method for Nonlinear First-Order Partial Differential Equations

Classical solutions of nonlinear first-order partial differential equations are approximated by solutions of quasilinear systems of difference equations. Sufficient conditions for the convergence of the method are given. The proof of the stability of the difference problem is based on the comparison method. Nonlinear estimates of the Perron type are assumed for increment functions.This new approach to the numerical solution of nonlinear equations is based on the method of quasilinearization for mixed problems. Numerical examples are given.