FINITE DIFFERENCE APPROXIMATIONS FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS

Abstract. Classical solutions of nonlinear partial differential equations are approximated in the paper 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 a comparison method. This new approach to the numerical solving of nonlinear equations is generated by a linearization method for initial problems. Numerical examples are given. 1. Difference systems corresponding to nonlinear equations. For any metric spaces X and Y we denote by C(X,Y) the class of all contin-uous functions from X into Y. We will use vectorial inequalities with the understanding that the same inequalities hold between their corresponding components. Let E be the Haar pyramid E = { (t, x) = (t, x1,..., xn) ∈ R1+n: t ∈ [0, a], −b+Mt ≤ x ≤ b−Mt} where a> 0, M = (M1,...,Mn) ∈ Rn+, R+ = [0,+∞), b = (b1,..., bn) ∈ R