An Energy-Preserving Wavelet Collocation Method for General Multi-Symplectic Formulations of Hamiltonian PDEs
Year: 2016
Communications in Computational Physics, Vol. 20 (2016), Iss. 5 : pp. 1313–1339
Abstract
In this paper, we develop a novel energy-preserving wavelet collocation method for solving general multi-symplectic formulations of Hamiltonian PDEs. Based on the autocorrelation functions of Daubechies compactly supported scaling functions, the wavelet collocation method is conducted for spatial discretization. The obtained semi-discrete system is shown to be a finite-dimensional Hamiltonian system, which has an energy conservation law. Then, the average vector field method is used for time integration, which leads to an energy-preserving method for multi-symplectic Hamiltonian PDEs. The proposed method is illustrated by the nonlinear Schrödinger equation and the Camassa-Holm equation. Since differentiation matrix obtained by the wavelet collocation method is a cyclic matrix, we can apply Fast Fourier transform to solve equations in numerical calculation. Numerical experiments show the high accuracy, effectiveness and conservation properties of the proposed method.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.231014.110416a
Communications in Computational Physics, Vol. 20 (2016), Iss. 5 : pp. 1313–1339
Published online: 2016-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 27
-
Line Integral Solution of Hamiltonian PDEs
Brugnano, Luigi | Frasca-Caccia, Gianluca | Iavernaro, FeliceMathematics, Vol. 7 (2019), Iss. 3 P.275
https://doi.org/10.3390/math7030275 [Citations: 13] -
Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation
Jiang, Chaolong | Wang, Yushun | Gong, YuezhengApplied Numerical Mathematics, Vol. 151 (2020), Iss. P.85
https://doi.org/10.1016/j.apnum.2019.12.016 [Citations: 25] -
Linear and Hamiltonian-conserving Fourier pseudo-spectral schemes for the Camassa–Holm equation
Hong, Qi | Gong, Yuezheng | Lv, ZhongquanApplied Mathematics and Computation, Vol. 346 (2019), Iss. P.86
https://doi.org/10.1016/j.amc.2018.10.043 [Citations: 3] -
Energy preserving model order reduction of the nonlinear Schrödinger equation
Karasözen, Bülent | Uzunca, MuratAdvances in Computational Mathematics, Vol. 44 (2018), Iss. 6 P.1769
https://doi.org/10.1007/s10444-018-9593-9 [Citations: 17] -
Energy-Preserving Algorithms for the Benjamin Equation
Song, Yifu | Wang, YushunJournal of Scientific Computing, Vol. 72 (2017), Iss. 2 P.605
https://doi.org/10.1007/s10915-017-0371-5 [Citations: 1] -
Local Discontinuous Galerkin Methods for the Two-Dimensional Camassa–Holm Equation
Ma, Tian | Xu, YanCommunications in Mathematics and Statistics, Vol. 6 (2018), Iss. 3 P.359
https://doi.org/10.1007/s40304-018-0140-2 [Citations: 1] -
A high-order linearly implicit energy-preserving Partitioned Runge-Kutta scheme for a class of nonlinear dispersive equations
Cui, Jin | Fu, YayunNetworks and Heterogeneous Media, Vol. 18 (2023), Iss. 1 P.399
https://doi.org/10.3934/nhm.2023016 [Citations: 0] -
A decoupled, linearly implicit and high-order structure-preserving scheme for Euler–Poincaré equations
Gao, Ruimin | Li, Dongfang | Mei, Ming | Zhao, DanMathematics and Computers in Simulation, Vol. 218 (2024), Iss. P.679
https://doi.org/10.1016/j.matcom.2023.12.009 [Citations: 2] -
A Linearly Implicit Structure-Preserving Scheme for the Camassa–Holm Equation Based on Multiple Scalar Auxiliary Variables Approach
Jiang, Chaolong | Gong, Yuezheng | Cai, Wenjun | Wang, YushunJournal of Scientific Computing, Vol. 83 (2020), Iss. 1
https://doi.org/10.1007/s10915-020-01201-4 [Citations: 18] -
Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions
Cai, Wenjun | Jiang, Chaolong | Wang, Yushun | Song, YongzhongJournal of Computational Physics, Vol. 395 (2019), Iss. P.166
https://doi.org/10.1016/j.jcp.2019.05.048 [Citations: 66] -
Linearly implicit local energy-preserving algorithm for a class of multi-symplectic Hamiltonian PDEs
Cai, Jiaxiang | Shen, BangyuComputational and Applied Mathematics, Vol. 41 (2022), Iss. 1
https://doi.org/10.1007/s40314-021-01740-y [Citations: 2]