Commun. Comput. Phys., 16 (2014), pp. 1323-1354.


Solving Two-Mode Shallow Water Equations Using Finite Volume Methods

Manuel Jesus Castro Diaz 1*, Yuanzhen Cheng 2, Alina Chertock 3, Alexander Kurganov 2

1 Dpto. de Analisis Matematico, Facultad de Ciencias, Universidad de Malaga, Campus de Teatinos, 29071 Malaga, Spain.
2 Mathematics Department, Tulane University, New Orleans, LA 70118, USA.
3 Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA.

Received 18 May 2013; Accepted (in revised version) 23 May 2014
Available online 29 August 2014
doi:10.4208/cicp.180513.230514a

Abstract

In this paper, we develop and study numerical methods for the two-mode shallow water equations recently proposed in [S. Stechmann, A. Majda, and B. Khouider, Theor. Comput. Fluid Dynamics, 22 (2008), pp. 407-432]. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches - two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme - and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method.

AMS subject classifications: 76M12, 65M08, 86-08, 86A10, 35L65, 35L67

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Key words: Two-mode shallow water equations, nonconservative products, conditional hyperbolicity, finite volume methods, central-upwind schemes, splitting methods, upwind schemes.

*Corresponding author.
Email: castro@anamat.cie.uma.es (M. J. Castro Diaz), ycheng5@tulane.edu (Y. Cheng), chertock@math.ncsu.edu (A. Chertock), kurganov@math.tulane.edu (A. Kurganov)
 

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