The von Neumann paradox in weak shock reflection

We present a numerical solution of the Euler equations of gas dynamics for a weak-shock Mach reflection in a half-space. In our numerical solutions, t he incident, reflected, and Mach shocks meet at a triple point, and there i s a supersonic patch behind the triple point, as proposed by Guderley. A th eoretical analysis supports the existence of an expansion fan at the triple point, in addition to the three shocks. This solution is in complete agree ment with the numerical solution of the unsteady transonic small-disturbanc e equations obtained by Hunter & Brio (2000), which provides an asymptotic description of a weak-shock Mach reflection. The supersonic patch is extrem ely small, and this work is the first time it has been resolved in a numeri cal solution of the Euler equations. The numerical solution uses six levels of grid refinement around the triple point. A delicate combination of nume rical techniques is required to minimize both the effects of numerical diff usion and the generation of numerical oscillations at grid interfaces and s hocks.