Abstract
A number of physical phenomena are described by nonlinear hyperbolic equations. Presence of discontinuous solutions motivates the necessity of development of reliable numerical methods based on the fundamental mathematical properties of hyperbolic systems. Construction of such methods for systems more complicated than the Euler gas dynamic equations requires the investigation of existence and uniqueness of the self-similar solutions to be used in the development of discontinuity-capturing high-resolution numerical methods. This frequently necessitates the study of the behavior of discontinuities under vanishing viscosity and dispersion. We discuss these problems in the application to the magnetohydrodynamic equations, nonlinear waves in elastic media, and electromagnetic wave propagation in magnetics.
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Kulikovskii, A.G., Pogorelov, N.V., Semenov, A.Y. (1999). Mathematical Aspects of Numerical Solution of Hyperbolic Systems. In: Jeltsch, R., Fey, M. (eds) Hyperbolic Problems: Theory, Numerics, Applications. International Series of Numerical Mathematics, vol 130. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8724-3_10
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DOI: https://doi.org/10.1007/978-3-0348-8724-3_10
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