Glimm’s scheme for systems with almost-planar interactions. (English) Zbl 0746.35022

Schochet, Steven

Commun. Partial Differ. Equations 16, No. 8-9, 1423-1440 (1991).

The Cauchy problem for a strictly-hyperbolic system of nonlinear conservation laws for which each field is either genuinely nonlinear or linearly degenerate is considered. The condition \([R_ i(u),R_ j(u)]=0\), \(\forall i,j,u\), where \(R_ j=r_ j\nabla u\) of Glimm’s existence theorem is relaxed to \(C^ k_{ij}({\tilde{u}})=0\), \(\forall i,j,k\), where \([R_ i(u),R_ j(u)]=\sum_ k C^ k_{ij}(u)R_ k(u)\). The proved existence theorem is examined under the above mentioned requirement for the Euler equations.

Reviewer: K.Zlateva (Russe)

MSC:

35L65	Hyperbolic conservation laws
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L80	Degenerate hyperbolic equations
65M99	Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99	Numerical methods for partial differential equations, boundary value problems

Keywords:

Cauchy problem; strictly-hyperbolic system; nonlinear conservation laws; genuinely nonlinear; linearly degenerate; Euler equations

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