High-order compact difference methods for solving two-dimensional nonlinear wave equations

Shuaikang Wang; Yunzhi Jiang; Yongbin Ge; Shuaikang Wang; Yunzhi Jiang; Yongbin Ge

doi:10.3934/era.2023159

Electronic Research Archive

2023, Volume 31, Issue 6: 3145-3168. doi: 10.3934/era.2023159

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Research article

High-order compact difference methods for solving two-dimensional nonlinear wave equations

1.
Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan, China
2.
Basic Courses Teaching and Research Department, Yingkou Institute of Technology, Yingkou, China

Received: 17 November 2022 Revised: 28 February 2023 Accepted: 06 March 2023 Published: 23 March 2023

Nonlinear wave equations are widely used in many areas of science and engineering. This paper proposes two high-order compact (HOC) difference schemes with convergence orders of $ O\left({{\tau ^4} + h_x^4 + h_y^4} \right) $ that can be used to solve nonlinear wave equations. The first scheme is a nonlinear compact difference scheme with three temporal levels. After calculating the second-order spatial derivatives of the previous time level using the Padé scheme, numerical solutions of the next time level are obtained through repeated iterations. The second scheme is a three-level linearized compact difference scheme. Unlike the first scheme, iterations are not required and it obtains numerical solutions through an explicit calculation. The two proposed schemes are applied to solutions of the coupled sine-Gordon equations. Finally, some numerical experiments are presented to confirm the effectiveness and accuracy of the proposed schemes.
Citation: Shuaikang Wang, Yunzhi Jiang, Yongbin Ge. High-order compact difference methods for solving two-dimensional nonlinear wave equations[J]. Electronic Research Archive, 2023, 31(6): 3145-3168. doi: 10.3934/era.2023159

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Abstract

Nonlinear wave equations are widely used in many areas of science and engineering. This paper proposes two high-order compact (HOC) difference schemes with convergence orders of $ O\left({{\tau ^4} + h_x^4 + h_y^4} \right) $ that can be used to solve nonlinear wave equations. The first scheme is a nonlinear compact difference scheme with three temporal levels. After calculating the second-order spatial derivatives of the previous time level using the Padé scheme, numerical solutions of the next time level are obtained through repeated iterations. The second scheme is a three-level linearized compact difference scheme. Unlike the first scheme, iterations are not required and it obtains numerical solutions through an explicit calculation. The two proposed schemes are applied to solutions of the coupled sine-Gordon equations. Finally, some numerical experiments are presented to confirm the effectiveness and accuracy of the proposed schemes.

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