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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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We discuss the results of a numerical simulation of the hydrodynamic Bondi–Hoyle accretion obtained on the basis of a high‐resolution numerical scheme with the proper entropy correction procedure necessary to avoid the ‘weak’ non‐monotonicity intrinsic in these schemes. Both the axisymmetric and plane cases are considered. The axisymmetric accretion problem turns out to have steady‐state solutions for all determining parameters. Steady‐state solutions for the plane accretion are found in certain cases even for sinusoidally perturbed and strongly non‐uniform winds. Non‐stationary phenomena do exist but they are not very violent and can sometimes be attributed as well to numerical as to physical reasons.

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