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Solving Two-Mode Shallow Water Equations Using Finite Volume Methods

Published online by Cambridge University Press:  03 June 2015

Manuel Jesús Castro Diaz ,
Yuanzhen Cheng ,
Alina Chertock  and
Alexander Kurganov
Manuel Jesús Castro Diaz*
Affiliation:
Dpto. de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Malaga, Spain
Yuanzhen Cheng*
Affiliation:
Mathematics Department, Tulane University, New Orleans, LA 70118, USA
Alina Chertock*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
Alexander Kurganov*
Affiliation:
Mathematics Department, Tulane University, New Orleans, LA 70118, USA
*
Corresponding author.Email:castro@anamat.cie.uma.es
Email:ycheng5@tulane.edu
Email:chertock@math.ncsu.edu
Email:kurganov@math.tulane.edu
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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.

Keywords

76M1265M0886-0886A1035L6535L67Two-mode shallow water equationsnonconservative productsconditional hyperbolicityfinite volume methodscentral-upwind schemessplitting methodsupwind schemes
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 5 , November 2014 , pp. 1323 - 1354
Copyright
Copyright © Global Science Press Limited 2014

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