Noh's problem

The third example is a classic test of W. Noh [#!Noh!#] for an ideal gas with $\gamma=5/3$. The initial density is 1, the initial pressure is 0, and the initial velocities are directed toward the origin with magnitude 1. The solution is an infinite strength circularly symmetric shock reflecting from the origin; the density behind the shock is 16 , the shock speed is 1/3 and ahead of the shock, that is for $\sqrt{x^2+y^2}>t/3$, the density is $(1+t/\sqrt{x^2+y^2})$. The computational domain is $0\leq x \leq 1$, $0\leq y \leq 1$. At the boundaries x=1 and y=1 we used the exact density as a function of time and radius together with the initial pressure and velocity. The grid size is 75 by 75.

The result of this example up to t=1 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

This is a difficult problem. The Lagrangian codes dealing with this problem suffer from a very large error in the density at the center. We must admit to being pleasantly surprised that the composite does as well as the figure shows. The central error is quite small, and just as satisfying is the maintenance of circular symmetry.

Animations

Contour plot (40 kB MPEG)

Surface plot (145 kB MPEG)

Contour plot (652 kB GIF) - this can play any browser

Surface plot (877 kB GIF) - this can play any browser