A Linearly-Fitted Conservative (Dissipative) Scheme for Efficiently Solving Conservative (Dissipative) Nonlinear Wave PDEs
Year: 2017
Author: Kai Liu, Xinyuan Wu, Wei Shi
Journal of Computational Mathematics, Vol. 35 (2017), Iss. 6 : pp. 780–800
Abstract
The extended discrete gradient method is an extension of traditional discrete gradient method, which is specially designed to solve oscillatory Hamiltonian systems efficiently while preserving their energy exactly. In this paper, based on the extended discrete gradient method, we present an efficient approach to devising novel schemes for numerically solving conservative (dissipative) nonlinear wave partial differential equations. The new scheme can preserve the energy exactly for conservative wave equations. With a minor remedy to the extended discrete gradient method, the new scheme is applicable to dissipative wave equations. Moreover, it can preserve the dissipation structure for the dissipative wave equation as well. Another important property of the new scheme is that it is linearly-fitted, which guarantees much fast convergence for the fixed-point iteration which is required by an energy-preserving integrator. The efficiency of the new scheme is demonstrated by some numerical examples.
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/jcm.1612-m2016-0604
Journal of Computational Mathematics, Vol. 35 (2017), Iss. 6 : pp. 780–800
Published online: 2017-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Conservative (dissipative) wave PDEs Structure-preserving algorithm Linearly-fitted Average Vector Field formula Sine-Gordon equation.
Author Details
-
Energy-preserving exponential integrators of arbitrarily high order for conservative or dissipative systems with highly oscillatory solutions
Mei, Lijie | Huang, Li | Wu, XinyuanJournal of Computational Physics, Vol. 442 (2021), Iss. P.110429
https://doi.org/10.1016/j.jcp.2021.110429 [Citations: 12] -
Long-time momentum and actions behaviour of energy-preserving methods for semi-linear wave equations via spatial spectral semi-discretisations
Wang, Bin | Wu, XinyuanAdvances in Computational Mathematics, Vol. 45 (2019), Iss. 5-6 P.2921
https://doi.org/10.1007/s10444-019-09719-4 [Citations: 9] -
Geometric Integrators for Differential Equations with Highly Oscillatory Solutions
Long-Time Momentum and Actions Behaviour of Energy-Preserving Methods for Wave Equations
Wu, Xinyuan | Wang, Bin2021
https://doi.org/10.1007/978-981-16-0147-7_14 [Citations: 0] -
Geometric Integrators for Differential Equations with Highly Oscillatory Solutions
Linearly-Fitted Conservative (Dissipative) Schemes for Nonlinear Wave Equations
Wu, Xinyuan | Wang, Bin2021
https://doi.org/10.1007/978-981-16-0147-7_8 [Citations: 0] -
Two Structure-Preserving Time Discretizations for Gradient Flows
Jüngel, Ansgar | Stefanelli, Ulisse | Trussardi, LaraApplied Mathematics & Optimization, Vol. 80 (2019), Iss. 3 P.733
https://doi.org/10.1007/s00245-019-09605-x [Citations: 3] -
The formulation and analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein–Gordon equations
Wang, Bin | Wu, XinyuanIMA Journal of Numerical Analysis, Vol. 39 (2019), Iss. 4 P.2016
https://doi.org/10.1093/imanum/dry047 [Citations: 40] -
An integral evolution formula of boundary value problem for wave equations
Fu, Ting | Zhang, Mingqian | Liu, KaiApplied Mathematics Letters, Vol. 116 (2021), Iss. P.107066
https://doi.org/10.1016/j.aml.2021.107066 [Citations: 0] -
Geometric Integrators for Differential Equations with Highly Oscillatory Solutions
Energy-Preserving Schemes for High-Dimensional Nonlinear KG Equations
Wu, Xinyuan | Wang, Bin2021
https://doi.org/10.1007/978-981-16-0147-7_9 [Citations: 0]