Abstract
This article focuses on the development and analysis of a high-order compact (HOC) alternating direction implicit (ADI) method for solving a two-dimensional (2D) coupled sine-Gordon equations, which combines fourth-order compact difference for the approximations of the second spatial derivatives, Crank–Nicolson (C–N) method for temporal discretization and the approximation factorization of difference operators. By the discrete energy analysis method, it is shown that their solutions converge to the exact solutions with an order of two in time and four in space in L2-, H1, and -norms. Meanwhile, it is pointed out that the present method can be slightly adapted to the numerical solutions of other types of sine-Gordon equations, for example, sine-Gordon equation, doubled sine-Gordon equation, and damped wave equations. Finally, numerical findings testify the computational efficiency of the algorithms.
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