Volume 9, Issue 2
New Conservative Finite Volume Element Schemes for the Modified Regularized Long Wave Equation

Jinliang Yan, Ming-Chih Lai, Zhilin Li & Zhiyue Zhang

Adv. Appl. Math. Mech., 9 (2017), pp. 250-271.

Published online: 2018-05

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  • Abstract

In this paper, we propose a new energy-preserving scheme and a new momentum-preserving scheme for the modified regularized long wave equation. The proposed schemes are designed by using the discrete variational derivative method and the finite volume element method. For comparison, we also propose a finite volume element scheme. The conservation properties of the proposed schemes are analyzed and we find that the energy-preserving scheme can precisely conserve the discrete total mass and total energy, the momentum-preserving scheme can precisely conserve the discrete total mass and total momentum, while the finite volume element scheme merely conserve the discrete total mass. We also analyze their linear stability property using the Von Neumann theory and find that the proposed schemes are unconditionally linear stable. Finally, we present some numerical examples to illustrate the effectiveness of the proposed schemes.

  • Keywords

Energy, momentum, mass, finite volume element method, MRLW equation.

  • AMS Subject Headings

65M99, 70H06, 74J35, 74S99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-9-250, author = {}, title = {New Conservative Finite Volume Element Schemes for the Modified Regularized Long Wave Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {9}, number = {2}, pages = {250--271}, abstract = {

In this paper, we propose a new energy-preserving scheme and a new momentum-preserving scheme for the modified regularized long wave equation. The proposed schemes are designed by using the discrete variational derivative method and the finite volume element method. For comparison, we also propose a finite volume element scheme. The conservation properties of the proposed schemes are analyzed and we find that the energy-preserving scheme can precisely conserve the discrete total mass and total energy, the momentum-preserving scheme can precisely conserve the discrete total mass and total momentum, while the finite volume element scheme merely conserve the discrete total mass. We also analyze their linear stability property using the Von Neumann theory and find that the proposed schemes are unconditionally linear stable. Finally, we present some numerical examples to illustrate the effectiveness of the proposed schemes.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2014.m888}, url = {http://global-sci.org/intro/article_detail/aamm/12147.html} }
TY - JOUR T1 - New Conservative Finite Volume Element Schemes for the Modified Regularized Long Wave Equation JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 250 EP - 271 PY - 2018 DA - 2018/05 SN - 9 DO - http://doi.org/10.4208/aamm.2014.m888 UR - https://global-sci.org/intro/article_detail/aamm/12147.html KW - Energy, momentum, mass, finite volume element method, MRLW equation. AB -

In this paper, we propose a new energy-preserving scheme and a new momentum-preserving scheme for the modified regularized long wave equation. The proposed schemes are designed by using the discrete variational derivative method and the finite volume element method. For comparison, we also propose a finite volume element scheme. The conservation properties of the proposed schemes are analyzed and we find that the energy-preserving scheme can precisely conserve the discrete total mass and total energy, the momentum-preserving scheme can precisely conserve the discrete total mass and total momentum, while the finite volume element scheme merely conserve the discrete total mass. We also analyze their linear stability property using the Von Neumann theory and find that the proposed schemes are unconditionally linear stable. Finally, we present some numerical examples to illustrate the effectiveness of the proposed schemes.

Jinliang Yan, Ming-Chih Lai, Zhilin Li & Zhiyue Zhang. (2020). New Conservative Finite Volume Element Schemes for the Modified Regularized Long Wave Equation. Advances in Applied Mathematics and Mechanics. 9 (2). 250-271. doi:10.4208/aamm.2014.m888
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