A Front Tracking Method for Transitional Shock Waves

Non{strictly-hyperbolic and mixed-type conservation laws pose a challenge to computational methods. Speciically, the computation of transitional shock waves, which are dependent on the parabolic regularization of the conservation laws, is sensitive to the numerical viscosity. In this paper, we propose a front tracking algorithm that treats transitional shock waves correctly. The algorithm includes the computation of saddle-to-saddle connections in planar dynamical systems using the multiple shooting method. We compare our results with those of commonly-used shock capturing schemes, including the Lax-Friedrichs, Lax-Wendroo, and higher-order Godunov schemes. 1. Introduction Discontinuities, such as shock waves and material interfaces, are a common and signiicant feature of solutions of systems of conservation laws. A discontinuous wave is, at each point, characterized by the limiting states on its two sides and by its normal propagation speed, which are related by jump conditions derived from the conservation laws. The discontinuity, however, is a mathematical idealization: physically there are regularization mechanisms that provide a discontinuous wave with a continuous internal structure on a small length scale 6]. Usually the internal structure can be ignored, but certain types of discontinuities are sensitively dependent on the regularization procedure, not only in their internal structure, but also in their propagation speeds and limiting states. These waves occur in applications such as combustion 4], three-phase ow through a porous medium 2, 20], and magneto-hydrodynamics 22]. It is important to be able to correctly simulate them using numerical methods. One type of regularization-sensitive wave, the transitional (or undercompressive) shock wave, has been the subject of recent studies of Riemann solutions in non{strictly-hyperbolic systems of conservation laws 18, 12]. A transitional shock wave diiers from a classical shock wave in that fewer characteristic families impinge on the wave. Therefore the dynamical information carried into the wave along characteristics is insuucient to determine the evolution of the wave. The missing information comes from the internal structure of the shock wave, which means that the internal structure strongly innuences wave propagation.