Admissible Wave Fans and the Riemann Problem

Abstract

The property of systems of conservation laws to be invariant under uniform stretching of the space-time coordinates induces the existence of self-similar solutions, which stay constant along straight-line rays emanating from some focal point in space-time. Such solutions depict a collection of waves converging to the focal point and interacting there to produce a jump discontinuity which is in turn resolved into an outgoing wave fan.

This chapter investigates the celebrated Riemann problem, whose object is the resolution of jump discontinuities into wave fans. A solution will be constructed in three different ways, namely: (a) by the classical method of piecing together elementary centered solutions encountered in earlier chapters, i.e., constant states, shocks joining constant states, and centered rarefaction waves bordered by constant states or contact discontinuities; (b) by minimizing the total entropy production of the outgoing wave fan; and (c) by a vanishing viscosity approach which employs time-dependent viscosity so that the resulting dissipative system is invariant under stretching of coordinates, just like the original hyperbolic system. A new type of discontinuity, called a delta shock, will emerge in the process.

The issue of admissibility of wave fans will be raised. In particular, it will be examined whether shocks contained in solutions constructed by any one of the above methods are necessarily admissible.

Next, the wave fan that best approximates the complex wave pattern generated by the interaction of two wave fans will be determined.

A system will be exhibited in which bounded initial data generate a resonating wave pattern that drives the solution amplitude to infinity, in finite time.

The chapter will close with a brief introduction to the theory of self-similar solutions to hyperbolic systems of conservation laws in two space dimensions.