Riemann problems for the two-dimensional unsteady transonic small disturbance equation

We study a two-parameter family of Riemann problems for the unsteady transonic small disturbance (UTSD) equation, also called the two-dimensional Burgers equation, which is used to model the transition from regular to Mach reflection for weak shock waves. The related initial-value problem consists of oblique shock data in the upper half-plane, with two parameters a and b corresponding to the slopes of the initial shock waves. The study of quasi-steady solutions leads to a problem that changes type when written in self-similar coordinates. The problem is hyperbolic in the region where the flow is supersonic, and elliptic where the flow is subsonic. In this paper we give a complete description of the flow in the hyperbolic region by resolving the hyperbolic wave interactions in the form of quasi-one-dimensional Riemann problems. In the region of physical space where the flow is subsonic, we pose the related free-boundary problems and discuss the behavior of the subsonic solution using results from our previous work. Based on this approach we establish the existence of regions of different qualitative behavior in parameter (a, b) space. Our results reveal that the UTSD equation seems to be particularly suitable for the study of the so-called von Neumann paradox in which linearly degenerate waves can be ignored. We establish the region in the parameter space where a prototype of von Neumann reflection takes place. In other regions of parameter space we find prototypes for Mach reflection, regular reflection, and transitional Mach reflection. The lack of linearly degenerate waves in this model is resolved by the presence of a small rarefaction wave emerging from the triple point.