Volume 27, Issue 2
Finite-Volume-Particle Methods for the Two-Component Camassa-Holm System

Alina Chertock, Alexander Kurganov & Yongle Liu

Commun. Comput. Phys., 27 (2020), pp. 480-502.

Published online: 2019-12

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  • Abstract

We study the two-component Camassa-Holm (2CH) equations as a model for the long time water wave propagation. Compared with the classical Saint-Venant system, it has the advantage of preserving the waves amplitude and shape for a long time. We present two different numerical methods—finite volume (FV) and hybrid finite-volume-particle (FVP) ones. In the FV setup, we rewrite the 2CH equations in a conservative form and numerically solve it by the central-upwind scheme, while in the FVP method, we apply the central-upwind scheme to the density equation only while solving the momentum and velocity equations by a deterministic particle method. Numerical examples are shown to verify the accuracy of both FV and FVP methods. The obtained results demonstrate that the FVP method outperforms the FV method and achieves a superior resolution thanks to a low-diffusive nature of a particle approximation.

  • Keywords

Two-component Camassa-Holm system, finite-volume method, deterministic particle method, finite-volume-particle method, central-upwind scheme.

  • AMS Subject Headings

65M08, 76M12, 76M28, 86-08, 76M25, 35L65

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

chertock@math.ncsu.edu (Alina Chertock)

alexander@sustech.edu.cn (Alexander Kurganov)

11749318@mail.sustech.edu.cn (Yongle Liu)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-27-480, author = {Chertock , Alina and Kurganov , Alexander and Liu , Yongle }, title = {Finite-Volume-Particle Methods for the Two-Component Camassa-Holm System}, journal = {Communications in Computational Physics}, year = {2019}, volume = {27}, number = {2}, pages = {480--502}, abstract = {

We study the two-component Camassa-Holm (2CH) equations as a model for the long time water wave propagation. Compared with the classical Saint-Venant system, it has the advantage of preserving the waves amplitude and shape for a long time. We present two different numerical methods—finite volume (FV) and hybrid finite-volume-particle (FVP) ones. In the FV setup, we rewrite the 2CH equations in a conservative form and numerically solve it by the central-upwind scheme, while in the FVP method, we apply the central-upwind scheme to the density equation only while solving the momentum and velocity equations by a deterministic particle method. Numerical examples are shown to verify the accuracy of both FV and FVP methods. The obtained results demonstrate that the FVP method outperforms the FV method and achieves a superior resolution thanks to a low-diffusive nature of a particle approximation.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0325}, url = {http://global-sci.org/intro/article_detail/cicp/13455.html} }
TY - JOUR T1 - Finite-Volume-Particle Methods for the Two-Component Camassa-Holm System AU - Chertock , Alina AU - Kurganov , Alexander AU - Liu , Yongle JO - Communications in Computational Physics VL - 2 SP - 480 EP - 502 PY - 2019 DA - 2019/12 SN - 27 DO - http://dor.org/10.4208/cicp.OA-2018-0325 UR - https://global-sci.org/intro/article_detail/cicp/13455.html KW - Two-component Camassa-Holm system, finite-volume method, deterministic particle method, finite-volume-particle method, central-upwind scheme. AB -

We study the two-component Camassa-Holm (2CH) equations as a model for the long time water wave propagation. Compared with the classical Saint-Venant system, it has the advantage of preserving the waves amplitude and shape for a long time. We present two different numerical methods—finite volume (FV) and hybrid finite-volume-particle (FVP) ones. In the FV setup, we rewrite the 2CH equations in a conservative form and numerically solve it by the central-upwind scheme, while in the FVP method, we apply the central-upwind scheme to the density equation only while solving the momentum and velocity equations by a deterministic particle method. Numerical examples are shown to verify the accuracy of both FV and FVP methods. The obtained results demonstrate that the FVP method outperforms the FV method and achieves a superior resolution thanks to a low-diffusive nature of a particle approximation.

Alina Chertock, Alexander Kurganov & Yongle Liu. (2019). Finite-Volume-Particle Methods for the Two-Component Camassa-Holm System. Communications in Computational Physics. 27 (2). 480-502. doi:10.4208/cicp.OA-2018-0325
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