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Energy-preserving Du Fort-Frankel difference schemes for solving sine-Gordon equation and coupled sine-Gordon equations

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Abstract

Du Fort-Frankel (DFF) finite difference method (FDM) was proposed for linear diffusion equations with periodic boundary conditions by Du Fort and Frankel in 1953. It is an explicit and unconditionally von Neumann stable scheme. Thus, it is very easy to be implemented and suitable for long-term simulations. However, there has been no research work on numerical solutions of sine-Gordon equations (SGE) and nonlinear coupled sine-Gordon equations (CSGEs) by using energy-preserving Du Fort-Frankel finite difference methods (EP-DFF-FDMs). In this study, two classes of weighted EP-DFF-FDMs, which are devised by combining DFF FDMs with invariant energy quadratization methods (IEQMs), are suggested for numerical simulations of SGE and CSGEs, respectively. By using the discrete energy method, it is shown that their solutions satisfy the discrete energy conservative laws, and converge to exact solutions with an order of \(\mathcal {O}(\tau ^{2}+{h^{2}_{x}}+{h^{2}_{y}}\) \(\displaystyle +(\frac {\tau }{h_{x}})^{2}+(\frac {\tau }{h_{y}})^{2})\) in H1-norm. Here, τ denotes time increment, while hx and hy represent spacing grids in x- and y-dimensions, respectively. What is more, our methods with parameter 𝜃 ≥ 1/4 are unconditionally stable in L2-norm though they are explicit schemes. Finally, numerical results confirm the exactness of theoretical findings, and the superiorities of our algorithms over some existent algorithms in terms of computational efficiency and the ability to conserve the discrete energy.

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Data Availability

We state that the datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors are very grateful to the referees for their valuable comments and suggestions, which have helped to improve the paper. We are also very grateful to Editor in Chief Claude Brezinski for his kind help.

Funding

This work is partly supported by the National Natural Science Foundation of China (Nos. 11861047), Natural Science Foundation of Jiangxi Province for Distinguished Young Scientists (No. 20212ACB211006), and Natural Science Foundation of Jiangxi province (No. 20202BABL201005).

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  1. College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang, 330063, China

    Dingwen Deng, Jingliang Chen & Qihong Wang

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  1. Dingwen Deng
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Correspondence to Dingwen Deng.

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Deng, D., Chen, J. & Wang, Q. Energy-preserving Du Fort-Frankel difference schemes for solving sine-Gordon equation and coupled sine-Gordon equations. Numer Algor 93, 1045–1081 (2023). https://doi.org/10.1007/s11075-022-01453-1

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