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Volume 4, Issue 4
New Finite-Volume Relaxation Methods for the Third-Order Differential Equations

Fayssal Benkhaldoun & Mohammed Seaïd

Commun. Comput. Phys., 4 (2008), pp. 820-837.

Published online: 2008-04

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  • Abstract

We propose a new method for numerical solution of the third-order differential equations. The key idea is to use relaxation approximation to transform the nonlinear third-order differential equation to a semilinear second-order differential system with a source term and a relaxation parameter. The relaxation system has linear characteristic variables and can be numerically solved without relying on Riemann problem solvers or linear iterations. A non-oscillatory finite volume method for the relaxation system is developed. The method is uniformly accurate for all relaxation rates. Numerical results are shown for some nonlinear problems such as the Korteweg-de Vires equation. Our method demonstrated the capability of accurately capturing soliton wave phenomena.

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@Article{CiCP-4-820, author = {}, title = {New Finite-Volume Relaxation Methods for the Third-Order Differential Equations}, journal = {Communications in Computational Physics}, year = {2008}, volume = {4}, number = {4}, pages = {820--837}, abstract = {

We propose a new method for numerical solution of the third-order differential equations. The key idea is to use relaxation approximation to transform the nonlinear third-order differential equation to a semilinear second-order differential system with a source term and a relaxation parameter. The relaxation system has linear characteristic variables and can be numerically solved without relying on Riemann problem solvers or linear iterations. A non-oscillatory finite volume method for the relaxation system is developed. The method is uniformly accurate for all relaxation rates. Numerical results are shown for some nonlinear problems such as the Korteweg-de Vires equation. Our method demonstrated the capability of accurately capturing soliton wave phenomena.

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7816.html} }
TY - JOUR T1 - New Finite-Volume Relaxation Methods for the Third-Order Differential Equations JO - Communications in Computational Physics VL - 4 SP - 820 EP - 837 PY - 2008 DA - 2008/04 SN - 4 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cicp/7816.html KW - AB -

We propose a new method for numerical solution of the third-order differential equations. The key idea is to use relaxation approximation to transform the nonlinear third-order differential equation to a semilinear second-order differential system with a source term and a relaxation parameter. The relaxation system has linear characteristic variables and can be numerically solved without relying on Riemann problem solvers or linear iterations. A non-oscillatory finite volume method for the relaxation system is developed. The method is uniformly accurate for all relaxation rates. Numerical results are shown for some nonlinear problems such as the Korteweg-de Vires equation. Our method demonstrated the capability of accurately capturing soliton wave phenomena.

Fayssal Benkhaldoun & Mohammed Seaïd. (2020). New Finite-Volume Relaxation Methods for the Third-Order Differential Equations. Communications in Computational Physics. 4 (4). 820-837. doi:
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