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The Glimm Difference Scheme

  • Joel Smoller
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 258)

Abstract

We consider a general system of conservation laws
$$ {{u}_{t}}{\text{ + }}f{{(u)}_{x}} = 0,\quad x \in R,\;\quad t > 0, $$
(19.1)
where u = (u 1,⋯,u n), with initial data
$$ u(x,0) = {{u}_{0}}(x),\quad x \in R. $$
(19.2)
The system (19.1) is assumed to be hyperbolic and genuinely nonlinear in each characteristic field, in some open set U ⊂ ℝn (see Definition 17.7). We let λ1(u) < ⋯ < λn(u) denote the eigenvalues of df(u). Concerning u o(x), we assume that T.V.(u 0) is sufficiently small, where by T.V.(-) we mean the total variation. With these assumptions, we shall show that the above problem has a solution which exists for all t > 0.

Keywords

Shock Wave Approximate Solution Difference Scheme Rarefaction Wave Difference Approximation 
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.

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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Joel Smoller
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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