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Volume 25, Issue 2
Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations

Qian Zhang & Yinhua Xia

DOI: 10.4208/cicp.OA-2017-0204

Commun. Comput. Phys., 25 (2019), pp. 532-563.

Published online: 2018-10

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  • 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.

  • Keywords

Local discontinuous Galerkin method, conservative and dissipative schemes, Korteweg-de Vries type equations, semi-implicit spectral deferred correction method.

  • AMS Subject Headings

65M60, 65M12, 35Q53

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-25-532, author = {Qian Zhang and Yinhua Xia}, title = {Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {25}, number = {2}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0204}, url = {http://global-sci.org/intro/article_detail/cicp/12762.html} }
TY - JOUR T1 - Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations AU - Qian Zhang & Yinhua Xia JO - Communications in Computational Physics VL - 2 SP - 532 EP - 563 PY - 2018 DA - 2018/10 SN - 25 DO - http://doi.org/10.4208/cicp.OA-2017-0204 UR - https://global-sci.org/intro/article_detail/cicp/12762.html KW - Local discontinuous Galerkin method, conservative and dissipative schemes, Korteweg-de Vries type equations, semi-implicit spectral deferred correction method. AB -

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.

Qian Zhang and Yinhua Xia. (2018). Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations. Communications in Computational Physics. 25 (2). 532-563. doi:10.4208/cicp.OA-2017-0204
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