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An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions

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Abstract

In this paper, a novel energy-preserving numerical scheme for nonlinear Hamiltonian wave equations with Neumann boundary conditions is proposed and analyzed based on the blend of spatial discretization by finite element method (FEM) and time discretization by Average Vector Field (AVF) approach. We first use the finite element discretization in space, which leads to a system of Hamiltonian ODEs whose Hamiltonian can be thought of as the semi-discrete energy of the original continuous system. The stability of the semi-discrete finite element scheme is analyzed. We then apply the AVF approach to the Hamiltonian ODEs to yield a new and efficient fully discrete scheme, which can preserve exactly (machine precision) the semi-discrete energy. The blend of FEM and AVF approach derives a new and efficient numerical scheme for nonlinear Hamiltonian wave equations. The numerical results on a single-soliton problem and a sine-Gordon equation are presented to demonstrate the remarkable energy-preserving property of the proposed numerical scheme.

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Authors and Affiliations

  1. College of Mathematical Sciences, Nanjing Tech University, Nanjing, People’s Republic of China

    Wei Shi

  2. College of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, People’s Republic of China

    Kai Liu

  3. Department of Mathematics, Nanjing University, Nanjing, People’s Republic of China

    Xinyuan Wu

  4. State Key Laboratory for Novel Software Technology at Nanjing University, Nanjing, People’s Republic of China

    Xinyuan Wu

  5. College of Math and Statistics, Nanjing University of Information Science and Technology, Nanjing, People’s Republic of China

    Changying Liu

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  1. Wei Shi
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  2. Kai Liu
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  3. Xinyuan Wu
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Correspondence to Changying Liu.

Additional information

The research was supported in part by the Natural Science Foundation of China under Grant 11501288 and 11671200, by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant 20100091110033, by the 985 Project at Nanjing University under Grant 9112020301, by A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, by the Natural Science Foundation of Jiangsu Province under Grant BK20150934, and by the Natural Science Foundation of the Jiangsu Higher Education Institutions under Grant 16KJB110010 and 14KJB110009.

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Shi, W., Liu, K., Wu, X. et al. An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions. Calcolo 54, 1379–1402 (2017). https://doi.org/10.1007/s10092-017-0232-5

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  • DOI: https://doi.org/10.1007/s10092-017-0232-5

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