Two-Dimensional Riemann Problems for the Compressible Euler System∗∗

Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult, but a viable alternative remains missing. We list merits of one-dimensional Riemann problems and compare them with those for the current two-dimensional Riemann problems, to illustrate their worthiness. We approach twodimensional Riemann problems via the methodology promoted by Andy Majda in the spirits of modern applied mathematics; that is, simplified model building via asymptotic analysis, numerical simulation, and theoretical analysis. We derive a simplified model, called the pressure gradient system, from the full Euler system via an asymptotic process. We use state-of-the-art numerical methods in numerical simulations to discern small-scale structures of the solutions, e.g., semi-hyperbolic patches. We use analytical methods to establish the validity of the structure revealed in the numerical simulation. The entire process, used in many of Majda’s programs, is shown here for the two-dimensional Riemann problems for the compressible Euler systems of conservation laws.