Numerical Simulation of the Coupled Sine-Gordon Equations via a Linearized and Decoupled Compact ADI Method

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 $L^{\infty }$-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.

Keywords:

• Convergence
• coupled sine-Gordon equations
• HOC ADI method
• solvability

MATHEMATICS SUBJECT CLASSIFICATION:

• 65M06; 65M12; 65M15.

Disclosure statement

No potential conflict of interest was reported by the authors.

Acknowledgments

The author gives sincere thanks to two anonymous referees and Principal Editor Prof. Zuhair Nashed for their valuable comments and suggestions, which have greatly improved this article.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11861047) and Natural Science Foundation of Jiangxi Provincial Education Department (Grant No. GJJ160706, DA201807160).