Two analytical solutions for the reflection of unsteady shock wave and relevant numerical tests

Abstract

We have been testing methods of Godunov and Glimm-Chorin,also FLIC method,by comparing numerical results vs. our analytical solution of high accuracy. The analytical solution satisfies the Euler's equations exactly. The Rankine-Hugoniot conditions along the shock is also satisfied with high accuracy. Our analytical solution is neither homoentropic, nor self-similar, besides,the intensity of shock wave varies with time. The figures in the text show the tendency of our analytical solution. Our tests demands more from the numerical method than those tests, which include shock wave only of constant intensity. By suitable arrangements, especially in choicing reasonable spatial steplength, we can make the numerical results rather accurate in the neighbourhood of the shock wave, and on the other hand, through these tests,we have observed the remaining small errors not only qualitatively but also quantitatively. The results of the test of other methods shall be presented elsewhere.

It is worth while to notice that from all the figures in this paper one can see that the numerical results obtained byany of the three methods coincide well with Sedov's solution, where it holds, even if large spatical steplength is used. It seems that self-similar solution is easier to compute by numerical methods.

For the methods of Godunov, of Glimm-Chorin, we reduce the computing effort by an iterative procedure in solving the Riemann problem, this is important if we generalize the methods to two dimensional unsteady flow. For the FLIC method we modify the artifical viscosity term on the basis of a numerical analysis considering both the two steps as a whole for the region of small particle velocity.