Accuracy improvement of a Predictor–Corrector compact difference scheme for the system of two-dimensional coupled nonlinear wave equations

Abstract

The nonlinear couple wave equations, which are extensively applied in scientific fields, such as, solid state physics, quantum mechanics, nonlinear optics, are a kind of important evolution equations. This paper is concerned with their numerical solutions via the combinations of compact difference method, Predictor–Corrector (P–C) iterative methods and Richardson extrapolation methods (REMs). Firstly, fourth-order compact difference methods are used to discrete temporal and spatial derivatives, thus forming a nonlinear fully discrete compact difference scheme. By utilizing the discrete energy analysis method and fixed point theorem, we can prove that under the condition of accepted stable criterion this scheme is conditionally convergent with an order of O(τ4+hx4+hy4) in H1-norm, and solvable. For avoiding solving the system of nonlinear algebraic equations, a P–C iterative method is introduced to save time cost and make the implementation simple. Besides, REMs are further applied to attain higher-order accurate approximate solutions. Finally, numerical findings confirm the exactness of theoretical results and efficiency of the algorithms.