The time fourth-order compact ADI methods for solving two-dimensional nonlinear wave equations

Nonlinear wave equation is extensively applied in a wide variety of scientific fields, such as nonlinear optics, solid state physics and quantum field theory. In this paper, two high-performance compact alternating direction implicit (ADI) methods are developed for the nonlinear wave equations. The first scheme is developed a three-level nonlinear difference scheme for nonlinear wave equations, where in x-direction, series of linear tridiagonal systems are solved by Thomas algorithm, while in y-direction, nonlinear algebraic system are computed by Newton's iterative method. In contrast, the second scheme is linear, and permits the multiple uses of the Thomas algorithm in both x- and y-directions, thus it saves much time cost. By using the discrete energy analysis method, it is shown that both the developed schemes can attain numerical accuracy of order O(τ4+hx4+hy4) in H1-norm. Meanwhile, by the fixed point theorem and symmetric positive-definite properties of coefficient matrix, it is proved that they are both uniquely solvable. Besides, the proposed schemes are extended to the numerical solutions of the coupled sine-Gordon wave equations and damped wave equations. Finally, numerical results confirm the convergence orders and exhibit efficiency of our algorithms.