Maximum error estimates of two linearized compact difference schemes for two-dimensional nonlinear Sobolev equations

Abstract

In this paper, two classes of high-order numerical schemes on the time discretization for the solutions of two-dimensional nonlinear Sobolev equations are analyzed. The two-level Newton linearized difference scheme had been established in the previous literature, in which the error estimates in L2-norm and H1-norm are obtained, but that in L∞-norm remains unsolved. The second difference scheme is established based on a three-level linearized difference technique in temporal direction. The numerical discretization in spatial dimension for both schemes utilizes the compact difference operator. We prove that the orders of convergence for both difference schemes in L∞-norm are O(τ2+h14+h24) based on the energy argument, where τ denotes the temporal step size, h1 and h2 denote the spatial step sizes in x-direction and y-direction, respectively. The numerical tests further verify the theoretical results and demonstrate the efficiency of both difference schemes.