Commun. Comput. Phys.,
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Volume 4.

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

Fayssal Benkhaldoun 1, Mohammed Seaid 2*

1 CMLA, ENS Cachan, 61 avenue du Pdt Wison, 94 235 Cachan, France; and LAGA, Universite Paris 13, 99 Av J.B. Clement, 93430 Villetaneuse, France.
2 School of Engineering, University of Durham, South Road, Durham DH1 3LE, UK.

Received 21 November 2007; Accepted (in revised version) 17 April 2008
Available online 28 May 2008


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.

AMS subject classifications: 5L30, 76M12, 35Q53

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Key words: Third-order differential equations, relaxation approximation, finite volume method, Korteweg-de Vries equation, solitons.

*Corresponding author.
Email: (F. Benkhaldoun), (M. Seaid)

The Global Science Journal