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Volume 28, Issue 3
Central Discontinuous Galerkin Methods for the Generalized Korteweg-de Vries Equation

Mengjiao Jiao, Yingda Cheng, Yong Liu & Mengping Zhang

DOI: 10.4208/cicp.OA-2019-0099

Commun. Comput. Phys., 28 (2020), pp. 927-966.

Published online: 2020-07

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  • Abstract

In this paper, we develop central discontinuous Galerkin (CDG) finite element methods for solving the generalized Korteweg-de Vries (KdV) equations in one dimension. Unlike traditional discontinuous Galerkin (DG) method, the CDG methods evolve two approximate solutions defined on overlapping cells and thus do not need numerical fluxes on the cell interfaces. Several CDG schemes are constructed, including the dissipative and non-dissipative versions. L2 error estimates are established for the linear and nonlinear equation using several projections for different parameter choices. Although we can not provide optimal a priori error estimate, numerical examples show that our scheme attains the optimal (k+1)-th order of accuracy when using piecewise k-th degree polynomials for many cases.

  • Keywords

Korteweg-de Vries equation, central DG method, stability, error estimates.

  • AMS Subject Headings

65M12, 65M15, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-28-927, author = {Jiao , MengjiaoCheng , YingdaLiu , Yong and Zhang , Mengping}, title = {Central Discontinuous Galerkin Methods for the Generalized Korteweg-de Vries Equation}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {3}, pages = {927--966}, abstract = {

In this paper, we develop central discontinuous Galerkin (CDG) finite element methods for solving the generalized Korteweg-de Vries (KdV) equations in one dimension. Unlike traditional discontinuous Galerkin (DG) method, the CDG methods evolve two approximate solutions defined on overlapping cells and thus do not need numerical fluxes on the cell interfaces. Several CDG schemes are constructed, including the dissipative and non-dissipative versions. L2 error estimates are established for the linear and nonlinear equation using several projections for different parameter choices. Although we can not provide optimal a priori error estimate, numerical examples show that our scheme attains the optimal (k+1)-th order of accuracy when using piecewise k-th degree polynomials for many cases.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0099}, url = {http://global-sci.org/intro/article_detail/cicp/17663.html} }
TY - JOUR T1 - Central Discontinuous Galerkin Methods for the Generalized Korteweg-de Vries Equation AU - Jiao , Mengjiao AU - Cheng , Yingda AU - Liu , Yong AU - Zhang , Mengping JO - Communications in Computational Physics VL - 3 SP - 927 EP - 966 PY - 2020 DA - 2020/07 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2019-0099 UR - https://global-sci.org/intro/article_detail/cicp/17663.html KW - Korteweg-de Vries equation, central DG method, stability, error estimates. AB -

In this paper, we develop central discontinuous Galerkin (CDG) finite element methods for solving the generalized Korteweg-de Vries (KdV) equations in one dimension. Unlike traditional discontinuous Galerkin (DG) method, the CDG methods evolve two approximate solutions defined on overlapping cells and thus do not need numerical fluxes on the cell interfaces. Several CDG schemes are constructed, including the dissipative and non-dissipative versions. L2 error estimates are established for the linear and nonlinear equation using several projections for different parameter choices. Although we can not provide optimal a priori error estimate, numerical examples show that our scheme attains the optimal (k+1)-th order of accuracy when using piecewise k-th degree polynomials for many cases.

Jiao , MengjiaoCheng , YingdaLiu , Yong and Zhang , Mengping. (2020). Central Discontinuous Galerkin Methods for the Generalized Korteweg-de Vries Equation. Communications in Computational Physics. 28 (3). 927-966. doi:10.4208/cicp.OA-2019-0099
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