Numerical computation of the Shock Tube Problem by means of wave digital principles

Abstract. Partial differential equations can be solved numerically by means of wave digital principles. The great advantage of this method is the simultaneous achievement of high robustness, massive parallelism full localness and high accuracy. Among others this method will be applied in order to solve the Euler-equations according to one dimension in space. Especially the so called Shock Tube Problem will be examined. The analytical solution of this problem contains two discontinuities, namely a shock and a contact discontinuity. These result in oscillations which are due to numerical integration methods of higher order. Also solutions of the Wave Digital Method contain these oscillations, contrary to what had been observed of Yuhui Zhu (2000). This behaviour is also known as Gibbs Phenomena.

The Navier-Stokes-equations, which are from a physical point of view more exactly, additionally take viscosity terms into account. This leads to smooth solutions near shocks. It will be shown that this approach leads to the suppression of the oscillations near the shock. Furthermore it will be shown that quite good results for the computation of velocity and pressure can be obtained.