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Approximate solutions of generalized Riemann problems for nonlinear systems of hyperbolic conservation laws


Authors: Claus R. Goetz and Armin Iske
Journal: Math. Comp. 85 (2016), 35-62
MSC (2010): Primary 65M08; Secondary 65D15, 58J45, 35L65
DOI: https://doi.org/10.1090/mcom/2970
Published electronically: May 13, 2015
MathSciNet review: 3404442
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Abstract: We study analytical properties of the Toro-Titarev solver for generalized Riemann problems (GRPs), which is the heart of the flux computation in ADER generalized Godunov schemes. In particular, we compare the Toro-Titarev solver with a local asymptotic expansion developed by LeFloch and Raviart. We show that for nonlinear scalar problems in 1D the Toro-Titarev solver reproduces the truncated Taylor series expansion of LeFloch-Raviart exactly, whereas for nonlinear systems the Toro-Titarev solver introduces an error whose size depends on the height of the jump in the initial data. Thereby, our analysis answers open questions concerning the justification of simplifying steps in the Toro-Titarev solver. We illustrate our results by giving the full analysis for a nonlinear $ 2$-by-$ 2$ system and numerical results for shallow water equations and a system from traffic flow.


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Additional Information

Claus R. Goetz
Affiliation: Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, I-38123 Trento, Italy
Email: clausruediger.goetz@unitn.it

Armin Iske
Affiliation: Department of Mathematics, University of Hamburg, Bundesstr. 55, D-20146 Hamburg, Germany
Email: armin.iske@uni-hamburg.de

DOI: https://doi.org/10.1090/mcom/2970
Keywords: Hyperbolic conservation laws, generalized Riemann problems, ADER methods
Received by editor(s): September 4, 2013
Received by editor(s) in revised form: April 24, 2014
Published electronically: May 13, 2015
Article copyright: © Copyright 2015 American Mathematical Society