Finite Difference Approximations for Nonlinear First Order Partial Differential Equations

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 continuous 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) ∈ R : t ∈ [0, a], −b+Mt ≤ x ≤ b−Mt } where a > 0, M = (M1, . . . ,Mn) ∈ Rn +, R+ = [0,+∞), b = (b1, . . . , bn) ∈ Rn and b ≥ Ma. Write Ω = E × R × Rn and suppose that f : Ω → R is a given function of the variables (t, x, p, q), q = (q1, . . . , qn). We consider the nonlinear first order partial differential equation (1) ∂tz(t, x) = f( t, x, z(t, x), ∂xz(t, x) ) with the initial condition (2) z(0, x) = φ(x), x ∈ [−b, b], where φ : [−b, b] → R is a given function and ∂xz = (∂x1z, . . . , ∂xnz). We are interested in the construction of a method for the approximation of solutions to problem (1), (2) with solutions of associated difference equations and in the