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On some fast well-balanced first order solvers for nonconservative systems


Authors: Manuel J. Castro, Alberto Pardo, Carlos Parés and E. F. Toro
Journal: Math. Comp. 79 (2010), 1427-1472
MSC (2000): Primary 74S10, 65M06, 35L60, 35L65, 35L67
Published electronically: November 23, 2009
MathSciNet review: 2629999
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Abstract: The goal of this article is to design robust and simple first order explicit solvers for one-dimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher order methods for one or multidimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good well-balanced and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different one-parameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods.


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Additional Information

Manuel J. Castro
Affiliation: Universidad de Málaga, Departamento Análisis Matemático, Campus de Teatinos s/n, 29071 Málaga, Spain
Email: castro@anamat.cie.uma.es

Alberto Pardo
Affiliation: Universidad de Málaga, Departamento Análisis Matemático, Campus de Teatinos s/n, 29071 Málaga, Spain
Email: pardo@anamat.cie.uma.es

Carlos Parés
Affiliation: Universidad de Málaga, Departamento Análisis Matemático, Campus de Teatinos s/n, 29071 Málaga, Spain
Email: pares@anamat.cie.uma.es

E. F. Toro
Affiliation: University of Trento. Laboratory of Applied Mathematics. Faculty of Engineering, 38050 Mesiano di Povo, Trento, Italy
Email: toroe@ing.unitn.it

DOI: https://doi.org/10.1090/S0025-5718-09-02317-5
Keywords: Nonconservative hyperbolic systems, finite volume method, approximate Riemann solvers, coefficient-splitting schemes, GFORCE method, well-balanced schemes, high order methods.
Received by editor(s): November 24, 2008
Received by editor(s) in revised form: May 11, 2009
Published electronically: November 23, 2009
Additional Notes: This research has been partially supported by the Spanish Government Research project MTM2006-08075. The numerical computations have been performed at the Laboratory of Numerical Methods of the University of Málaga.
Article copyright: © Copyright 2009 American Mathematical Society