The error estimations of a two-level linearized compact ADI method for solving the nonlinear coupled wave equations

In this article, a two-level linearized compact alternating direction implicit (ADI) scheme is proposed for solving two-dimensional (2D) nonlinear coupled wave equations (NCWEqs). In comparison with the existent compact ADI methods for NCWEqs, there are two obvious characters. (1) Numerical solutions at time level one, which have effect on the global stability and convergence of the numerical solutions, are not needed to be solved firstly by using another numerical scheme. (2) The computational cost is comparatively low and acceptable because the new compact ADI method is a linear scheme, thus avoiding Newton’s iterations. By using the energy analysis method, it is shown that numerical solutions converge to exact solutions with an order of O ( Δ t 2 + h x 4 + h y 4 ) in H 1 - and L ∞ -norms, and are uniquely solvable. Numerical examples are given to illustrate the exactness of the theoretical findings and the high-performance of our methods.