Advertisement

The Front Tracking Method and Standard Riemann Semigroups

  • Constantine M. DafermosEmail author
Chapter
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 325)

Abstract

A method is described in this chapter for constructing solutions of the initial value problem for hyperbolic systems of conservation laws by tracking the waves and monitoring their interactions as they collide. Interactions between shocks are easily resolved by solving Riemann problems; this is not the case, however, with interactions involving rarefaction waves. The random choice method, expounded in Chapter XIII, side-steps this difficulty by stopping the clock before the onset of wave collisions and reapproximating the solution by step functions. In contrast, the front tracking approach circumvents the obstacle by disposing of rarefaction waves altogether and resolving all Riemann problems in terms of shocks only. Such solutions generally violate the admissibility criteria. Nevertheless, considering the close local proximity between shock and rarefaction wave curves in state space, any rarefaction wave may be approximated arbitrarily close by fans of (inadmissible) shocks of very small strength. The expectation is that in the limit, as this approximation becomes finer, one recovers admissible solutions. The implementation of the front tracking algorithm, with proof that it converges, will be presented here, first for scalar conservation laws and then in the context of genuinely nonlinear strictly hyperbolic systems of conservation laws of any size.

Keywords

Rarefaction Wave Riemann Problem Jump Discontinuity Admissible Solution Riemann Solver 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations