A novel high-order symmetric and energy-preserving continuous-stage Runge-Kutta-Nyström Fourier pseudo-spectral scheme for solving the two-dimensional nonlinear wave equation

Dongjie Gao; Peiguo Zhang; Longqin Wang; Zhenlong Dai; Yonglei Fang; Dongjie Gao; Peiguo Zhang; Longqin Wang; Zhenlong Dai; Yonglei Fang

doi:10.3934/math.2025310

AIMS Mathematics

2025, Volume 10, Issue 3: 6764-6787. doi: 10.3934/math.2025310

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Research article

A novel high-order symmetric and energy-preserving continuous-stage Runge-Kutta-Nyström Fourier pseudo-spectral scheme for solving the two-dimensional nonlinear wave equation

1.
School of Mathematics and Statistics, Heze University, Heze 274015, China
2.
School of Statistics and Data Science, Jiangsu normal University, Xuzhou 221116, China
3.
School of Information Engineering, Nanjing Xiaozhuang University, Nanjing 211171, China
4.
School of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, China

Received: 17 January 2025 Revised: 14 March 2025 Accepted: 19 March 2025 Published: 26 March 2025
MSC : 34C14, 65L06, 65L20, 65L70

The primary objective of this research is to develop a novel high-order symmetric and energy-preserving method for solving two-dimensional nonlinear wave equations. Initially, the nonlinear wave equation is reformulated as an abstract Hamiltonian ordinary differential equation (ODE) system with separable energy in an appropriate infinite-dimensional function space. Subsequently, an energy-preserving and symmetric continuous-stage Runge-Kutta-Nyström time-stepping scheme is derived. After approximating the spatial differential operator using the two-dimensional Fourier pseudo-spectral method, we derive an energy-preserving fully discrete scheme. A rigorous error analysis demonstrates that the proposed method can achieve at least fourth-order accuracy in time. Finally, numerical examples are provided to validate the accuracy, efficiency, and long-term energy conservation of the method.
- two dimensional nonlinear wave equations,
- energy-preserving method,
- symmetry,
- continuous-stage Runge-Kutta-Nyström method,
- Fourier pseudo-spectral method
Citation: Dongjie Gao, Peiguo Zhang, Longqin Wang, Zhenlong Dai, Yonglei Fang. A novel high-order symmetric and energy-preserving continuous-stage Runge-Kutta-Nyström Fourier pseudo-spectral scheme for solving the two-dimensional nonlinear wave equation[J]. AIMS Mathematics, 2025, 10(3): 6764-6787. doi: 10.3934/math.2025310

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Abstract

The primary objective of this research is to develop a novel high-order symmetric and energy-preserving method for solving two-dimensional nonlinear wave equations. Initially, the nonlinear wave equation is reformulated as an abstract Hamiltonian ordinary differential equation (ODE) system with separable energy in an appropriate infinite-dimensional function space. Subsequently, an energy-preserving and symmetric continuous-stage Runge-Kutta-Nyström time-stepping scheme is derived. After approximating the spatial differential operator using the two-dimensional Fourier pseudo-spectral method, we derive an energy-preserving fully discrete scheme. A rigorous error analysis demonstrates that the proposed method can achieve at least fourth-order accuracy in time. Finally, numerical examples are provided to validate the accuracy, efficiency, and long-term energy conservation of the method.

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