ABSTRACT
We revisit and give a more detailed presentation of [3] by C˘anic´, Keyfitz, and Kim on a solution to a special Riemann problem for the unsteady transonic small disturbance (UTSD) equation. The Riemann initial data consist of two states in the upper half plane {(x, y) :x ∈ R, y ≥ 0} separated by an incident shock, and results in a regular reflection where the flow behind the reflected shock is subsonic. Written in self-similar coordinates ξ = x/t and η = y/t, this configuration leads to a system that changes type. We find a solution in the hyperbolic part of the domain using the standard theory of one-dimensional conservation laws and the notion of quasi-onedimensional Riemann problems developed in [1]. Solution in the elliptic part of the domain is described by a free boundary value problem. The free boundary is given by the position of the reflected shock which is, through the Rankine-Hugoniot relations, coupled to the subsonic state behind the shock. The main idea in solving this free boundary value problem is to fix the position of the reflected shock within some bounded set of admissible curves, solve the fixed boundary value problem, and then update the position of the reflected shock using the Rankine-Hugoniot relations.
