The Glimm Difference Scheme

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


We consider a general system of conservation laws
$$ {{u}_{t}}{\text{ + }}f{{(u)}_{x}} = 0,\quad x \in R,\;\quad t > 0, $$
where u = (u 1,⋯,u n), with initial data
$$ u(x,0) = {{u}_{0}}(x),\quad x \in R. $$
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.


Shock Wave Approximate Solution Difference Scheme Rarefaction Wave Difference Approximation 
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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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