Conservative second-order finite difference method for Camassa–Holm equation with periodic boundary condition

Abstract

In this paper, we propose two momentum-preserving finite difference schemes for solving one-dimensional Camassa–Holm equation with periodic boundary conditions. A two-level nonlinear difference scheme and a three-level linearized difference scheme are constructed by using the method of order reduction. For nonlinear scheme, we combine mid-point rule and a specific difference operator, which ensures that our obtained scheme is of second-order convergence in both temporal and spatial directions. For linearized scheme, we apply a linear implicit Crank–Nicolson scheme in the temporal direction, then unique solvability and momentum conservation are analysed in detail. Numerical experiments are provided for Camassa–Holm equation admitting different types of solutions, which demonstrate the convergence order and accuracy of the proposed methods coincide with theoretical analysis. Moreover, numerical results show that the nonlinear scheme exhibits better accuracy for mass conservation, while the linearized scheme is more time-saving in computation.

Keywords:

• Camassa–Holm equation
• invariant-preserving
• order reduction
• finite difference method
• error estimate

Mathematics Subject Classifications:

• 76M20
• 65M06
• 65M22
• 65J15
• 49M37

Acknowledgments

The authors would like to express their appreciation to Professor Zhilin Li (NC State University, USA), Professor Qin Sheng (Baylor University, USA) and Professor Hai-Wei Sun (University of Macao, Macau, P.R. China) for their kind help and continuous encouragement in numerical methods for PDEs.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partly supported by the National Natural Science Foundation of China (Grant No. 51974377), Natural Science Foundation of Hunan Province (Grant No. 2019JJ50755), and Postgraduate Research and Innovation Program of Central South University (Grant Nos. 2023ZZTS0690 and 2022ZZTS0607).