The Triple Point Paradox for the Nonlinear Wave System

Abstract

We present numerical solutions of a two-dimensional Riemann problem for the non- linear wave system which is used to describe the Mach reflection of weak shock waves. Robust low order as well as high resolution finite volume schemes are employed to solve this equation formulated in self-similar variables. These, together with extreme local grid refinement, are used to resolve the solution in the neighborhood of an apparent but mathematically inadmissible shock triple point. Rather than observing three shocks meeting in a single standard triple point, we are able to detect a primary triple point containing an additional wave, a centered expansion fan, together with a se- quence of secondary triple points and tiny supersonic patches embedded within the subsonic region directly behind the Mach stem. An expansion fan originates at each triple point. It is our opinion that the structure observed here resolves the von Neumann triple point paradox for the nonlinear wave system. These solutions closely resemble the solutions obtained in (A. M. Tesdall and J. K. Hunter, SIAM J. Appl. Math., 63 (2002), pp. 42-61) for the unsteady transonic small disturbance equation.