Self-Similar Solutions for the Triple Point Paradox in Gasdynamics

Abstract

We present numerical solutions of a two-dimensional Riemann problem for the compressible Euler equations that describes the Mach reflection of weak shock waves. High resolution finite volume schemes are used to solve the equations formulated in self-similar variables. We use extreme local grid refinement to resolve the solution in the neighborhood of an apparent but mathematically inadmissible shock triple point. The solutions contain a complex structure: instead of three shocks meeting in a single standard triple point, there is a sequence of triple points and tiny supersonic patches behind the leading triple point, formed by the reflection of weak shocks and expansion waves between the sonic line and the Mach shock. An expansion fan originates at each triple point, resolving the von Neumann triple point paradox.