Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations
Year: 2019
Communications in Computational Physics, Vol. 25 (2019), Iss. 2 : pp. 532–563
Abstract
In this paper, we develop the Hamiltonian conservative and $L^2$ conservative local discontinuous Galerkin (LDG) schemes for the Korteweg-de Vries (KdV) type equations with the minimal stencil. For the time discretization, we adopt the semi-implicit spectral deferred correction (SDC) method to achieve the high order accuracy and efficiency. Also we compare the schemes with the dissipative LDG scheme. Stability of the fully discrete schemes is provided by Fourier analysis for the linearized KdV equation. Numerical examples are shown to illustrate the capability of these schemes. Compared with the dissipative LDG scheme, the numerical simulations also indicate that the conservative LDG scheme with high order time discretization can reduce the long time phase error validly.
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.OA-2017-0204
Communications in Computational Physics, Vol. 25 (2019), Iss. 2 : pp. 532–563
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 32
Keywords: Local discontinuous Galerkin method conservative and dissipative schemes Korteweg-de Vries type equations semi-implicit spectral deferred correction method.
-
A Class of Efficient Hamiltonian Conservative Spectral Methods for Korteweg-de Vries Equations
Yin, Xu | Cao, WaixiangJournal of Scientific Computing, Vol. 94 (2023), Iss. 1
https://doi.org/10.1007/s10915-022-02061-w [Citations: 2] -
Discontinuous Galerkin methods for short pulse type equations via hodograph transformations
Zhang, Qian | Xia, YinhuaJournal of Computational Physics, Vol. 399 (2019), Iss. P.108928
https://doi.org/10.1016/j.jcp.2019.108928 [Citations: 4] -
A New Conservative Discontinuous Galerkin Method via Implicit Penalization for the Generalized Korteweg–de Vries Equation
Chen, Yanlai | Dong, Bo | Pereira, RebeccaSIAM Journal on Numerical Analysis, Vol. 60 (2022), Iss. 6 P.3078
https://doi.org/10.1137/22M1470827 [Citations: 4] -
Conservative discontinuous Galerkin methods for the nonlinear Serre equations
Zhao, Jianli | Zhang, Qian | Yang, Yang | Xia, YinhuaJournal of Computational Physics, Vol. 421 (2020), Iss. P.109729
https://doi.org/10.1016/j.jcp.2020.109729 [Citations: 0] -
Analysis of the error of approximation of two-layer difference schemes for the Korteweg de Vries equation
Bykovskaya, Elena Nikolaevna | Shapranov, Alexander Viktorovich | Mazhukin, Vladimir IvanovichKeldysh Institute Preprints, Vol. (2021), Iss. 1 P.1
https://doi.org/10.20948/prepr-2021-1 [Citations: 1] -
An efficient dissipation–preserving Legendre–Galerkin spectral method for the Higgs boson equation in the de Sitter spacetime universe
Zaky, Mahmoud A. | Hendy, Ahmed S.Applied Numerical Mathematics, Vol. 160 (2021), Iss. P.281
https://doi.org/10.1016/j.apnum.2020.10.013 [Citations: 24] -
Comparison of different discontinuous Galerkin methods based on various reformulations for gKdV equation: Soliton dynamics and blowup
Hong, Xue | Wei, Qianrui | Zhao, XiaofeiComputer Physics Communications, Vol. 300 (2024), Iss. P.109180
https://doi.org/10.1016/j.cpc.2024.109180 [Citations: 0] -
Analysis of Local Discontinuous Galerkin Methods with Implicit-Explicit Time Marching for Linearized KdV Equations
Wang, Haijin | Tao, Qi | Shu, Chi-Wang | Zhang, QiangSIAM Journal on Numerical Analysis, Vol. 62 (2024), Iss. 5 P.2222
https://doi.org/10.1137/24M1635818 [Citations: 0] -
Invariants Preserving Time-Implicit Local Discontinuous Galerkin Schemes for High-Order Nonlinear Wave Equations
Zheng, Wei | Xu, YanCommunications on Applied Mathematics and Computation, Vol. (2024), Iss.
https://doi.org/10.1007/s42967-024-00420-y [Citations: 0] -
A conservative local discontinuous Galerkin method with spectral accuracy for the Degasperis-Procesi equation
Song, Yifu | Zhu, MinDiscrete and Continuous Dynamical Systems - S, Vol. 0 (2024), Iss. 0 P.0
https://doi.org/10.3934/dcdss.2024172 [Citations: 0] -
Local Discontinuous Galerkin Methods to a Dispersive System of KdV-Type Equations
Zhang, Chao | Xu, Yan | Xia, YinhuaJournal of Scientific Computing, Vol. 86 (2021), Iss. 1
https://doi.org/10.1007/s10915-020-01370-2 [Citations: 3]