DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 1, Number 4, October 1995 pp. 555–584 ON THE 2-D RIEMANN PROBLEM FOR THE COMPRESSIBLE EULER EQUATIONS I. INTERACTION OF SHOCKS AND RAREFACTION WAVES Tung Chang (Tong Zhang) Institute of Mathematics, Academia Sinica, PRC Gui-Qiang Chen Department of Mathematics, Northwestern University, USA Shuli Yang Institute of Applied Mathematics, Academia Sinica, PRC Abstract. We are concerned with the Riemann problem for the two-dimensional com- pressible Euler equations in gas dynamics. The central point at this issue is the dynamical interaction of shock waves, centered rarefaction waves, and contact discontinuities that con- nect two neighboring constant initial states in the quadrants. The Riemann problem is classified into eighteen genuinely different cases. For each configuration, the structure of the Riemann solution is analyzed using the method of characteristics, and corresponding numerical solution is illustrated by contour plots using an upwind averaging scheme that is second order in the smooth region of the solution. In the first paper we mainly focus on the interaction of shocks and rarefaction waves. The theory is developed from an analysis of the structure of the Euler equations and their Riemann solutions in [CC, ZZ] and the MmB scheme [WY]. 1. Introduction We are concerned with the two-dimensional Riemann problem for gas dynamics. The Riemann problem, the fundamental block for the whole building of nonlinear conservation laws, has been the subject of many reviews (cf. [Da, Gli1, GM, La1]) and monographs (cf. [CH, CF, La2, Lu1]). The study of this problem has an extensive history dating back to the pioneer work of Riemann [Ri] in 1858. For one-dimensional case, a theory has been established for appropriate amplitude of the Riemann data (cf. [La2, Lu1]) for strictly hyperbolic systems and for general Riemann data (cf. [CH, MP, Sm, We] and the references cited there) for the compressible Euler equations. This problem is much more complicated for multidimensional case. Theoretical results are available only for scalar conservation laws [Gu, Li, Wa, ZZ1, CT] and a special 2 × 2 system of conservation laws [TZ]. The Riemann problem plays an essential role in developing one-dimensional the- ory of hyperbolic conservation laws (cf. [GM, GR, La2, Le, Sm]). It is the simplest one of general Cauchy problems and is much easier to clarify the explicit structure of its solution. On the other hand, the solution of the Cauchy problems can be locally approached by the solution of the Riemann problem (cf. [Ch, Gli2, Lu1]). Based on these features of this problem, the Riemann solution serves as a building AMS(MOS) Subject Classifications. Primary: 65M06, 76N10, 35L65, 35L67; Secondary: 65M99 Key Words: 2-D Riemann problem, Euler equations, transonic flow, shocks, rarefaction waves, slip curves, vortices, interaction of waves, upwind averaging schemes. 555