Two-Dimensional Riemann Problems for Ideal Gas Dynamics

The compressible Euler equations describing the flow of an ideal gas is a particular example of hyperbolic systems. Since the mathematical structure of the solution to the Euler equations is well-understood, this system serves as the main benchmark for testing new numerical methods developed for fluid dynamics problems with a similar mathematical character.

The Riemann problem is the simplest possible initial value problem for hyperbolic systems. In one spatial dimension, the Riemann (or shock-tube) problem is composed of two uniform states in the infinite domain separated by a discontinuity at the origin. For the Euler equations the exact solution of the Riemann problem is well-known, self-similar, and consists of a combination of three wave types: shocks (S), rarefaction waves (R), and contact discontiunitites (R). Apart from being an important testbench, the Riemann problem is a basic building block for a large class of modern numerical methods, called upwind or Godunov schemes.

In two-spatial dimensions, the Riemann problems consists of four uniform states, one in each quadrant. Compared with the relatively simple wave patterns in the 1-D case, the 2-D case gives rise to quite complicated wave patterns. These patterns can be characterised as 19 different configurations (see Schulz-Rinne, Collins and Glaz, SIAM J. Sci. Comp., Vol 14, No. 6, 1993), as shown in the images (and movies) below. The solutions are computed using a second-order, staggered, nonoscillatory, central difference scheme. The same 19 cases have been studied previously by Kurganov and Tadmor using related central difference schemes. The solutions are plotted as emulated Schlieren images by depicting the norm of the density gradient in a nonlinear graymap. This visualisation technique is particulary useful for depicting gradients in the density, but is also excellent for enhancing small-scale variances, for instance, the small numerical artifacts seen in several of the images below (pairs of horisontal and vertical lines).