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Effects of viscosity on a shock wave solution of the Euler equations. (English) Zbl 1117.76031
Hou, Thomas Y. (ed.) et al., Hyperbolic problems: Theory, numerics, applications. Proceedings of the ninth international conference on hyperbolic problems, Pasadena, CA, USA, March 25–29, 2002. Berlin: Springer (ISBN 3-540-44333-9/hbk). 655-664 (2003).
The authors consider Euler equations with an equation of state admitting multiple solutions to Riemann problems. Earlier several types of instabilities for this problem were investigated numerically using Lax-Friedrichs scheme on a staggered grid. Here the authors re-examine one case, a Riemann problem where the end states can be connected with a 3-shock or with two shock waves and an intermediate state. With discontinuous initial data the Lax-Friedrichs scheme produces the solution for the intermediate state. Further computations show that with sufficiently smooth initial data the Lax-Friedrichs scheme produces the discrete 3-shock wave solution. Similar results are obtained with other schemes. However, they note that the smoothness requirement (in order to obtain a 3-shock solution) of the initial profile is dependent on the numerical method used.
To understand this behaviour, the authors investigate the stability of a corresponding viscous problem. They find that the travelling wave solution of the viscous problem, corresponding to the 3-shock solution, satisfies the stability condition of Kreiss and Kreiss. Numerical computations never yield the 3-shock solution. The first hypothesis of the authors is that the 3-shock is unstable. Since numerical viscosity is present in the computations, they analyse the corresponding viscous problem. The authors find that sufficient conditions for nonlinear stability are satisfied.
In another set of experiments the authors use smoother initial data. Then a discrete 3-shock develops. The smoothness requirement is related to the steepness of the discrete shock profile, and thus it depends on the numerical method. Since reality is viscous, it might not be realistic to use discontinuous initial data. However, if the physical viscosity is much smaller than the numerical viscosity, using a physical shock as initial data might yield a wrong answer.
For the entire collection see [Zbl 1024.00068].

76L05 Shock waves and blast waves in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations