zbMATH — the first resource for mathematics

Glimm’s scheme for systems with almost-planar interactions. (English) Zbl 0746.35022
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
Full Text:
References:
 [1] Courant, R. and Friedrichs, K. O. 1948. ”Supersonic Flow and Shock Waves”. New York: springer–Verlag. · Zbl 0041.11302 [2] DOI: 10.1002/cpa.3160180408 · Zbl 0141.28902 · doi:10.1002/cpa.3160180408 [3] 1973.Hyperbolic systes of Conservation Laws and the Mathematical Theory of Shock Waves, 28 [4] DOI: 10.1512/iumj.1977.26.26011 · Zbl 0361.35056 · doi:10.1512/iumj.1977.26.26011 [5] Smoller, J. 1983.Shock Waves and Reaction–Diffusion Equaitons, 917–924. New york: Springer–Verlag. [6] DOI: 10.1016/0022-0396(81)90055-3 · Zbl 0476.76070 · doi:10.1016/0022-0396(81)90055-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.