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Mathematical aspects of numerical solution of hyperbolic systems. (English) Zbl 0926.35011
Fey, Michael (ed.) et al., Hyperbolic problems: Theory, numerics, applications. Proceedings of the 7th international conference, Zürich, Switzerland, February 1998. Vol. II. Basel: Birkhäuser. ISNM, Int. Ser. Numer. Math. 130, 589-598 (1999).
In the paper the authors describe some problems arising if numerical methods, which work well for the gas dynamics system of the Euler equations, are applied to the more general nonlinear hyperbolic systems, such as the magnetohydrodynamic equations (MHD), shallow water equations, elasticity equations, etc. Presence of discontinuities in the solution requires reliable numerical methods based on the fundamental mathematical properties of hyperbolic systems. Construction of such numerical methods requires the investigation of existence and uniqueness of the self-similar solutions, which are used in the development of shock-capturing high-resolution numerical methods. Therefore it is necessary to study discontinuities under vanishing viscosity and dispersion. The problems mentioned above are discussed in the paper for MHD equations, nonlinear waves in elastic media and electromagnetic wave propagation in magnetics.
For the entire collection see [Zbl 0911.00029].

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
35L60 First-order nonlinear hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
76M25 Other numerical methods (fluid mechanics) (MSC2010)