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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

MR Author ID:
600018

Email:
armin.iske@uni-hamburg.de

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