Gas Dynamics: The Riemann Problem and Discontinuous Solutions: Application to the Shock Tube Problem

Abstract

The interest in studying the shock tube problem is threefold. From a fundamental point of view, it offers an interesting framework to introduce some basic notions about nonlinear hyperbolic systems of partial differential equations (PDEs). From a numerical point of view, this problem constitutes, since the exact solution is known, an inevitable and difficult test case for any numerical method dealing with discontinuous solutions. Finally, there is a practical interest, since this model is used to describe real shock tube experimental devices.¹

The first shock tube facility was built in 1899 by Paul Vieille to study the deflagration of explosive charges. Nowadays, shock tubes are currently used as low-cost high-speed wind tunnels, in which a wide variety of aerodynamic or aeroballistic topics are studied: supersonic aircraft flight, gun performance, asteroid impacts, shuttle atmospheric entry, etc.