Abstract. The adaptive generalized Riemann problem (GRP) scheme for 2-D compressible fluid flows has been proposed in [25, Han, Li and Tang, An adaptive GRP scheme for compressible fluid flows, J. Comput. Phys., 229 (2010Phys., 229 ( ), 1448Phys., 229 ( -1466 and it displays the capability in overcoming difficulties such as the start-up error for a single shock, and the numerical instability of the almost stationary shock. In this paper, we will provide the accuracy study and particularly show the performance in simulating 2-D complex wave configurations formulated with the 2-D Riemann problems for compressible Euler equations. For this purpose, we will first review the GRP scheme briefly when combined with the adaptive moving mesh technique and consider the accuracy of the adaptive GRP scheme via the comparison with the explicit formulae of analytic solutions of planar rarefaction waves, planar shock waves, the collapse problem of a wedge-shaped dam and the spiral formation problem. Then we simulate the full set of wave configurations in the 2-D four-wave Riemann problems for compressible Euler equations [60, Zhang and Zheng, Conjecture on the structure of solutions of the Riemann problem for 2-D gas dynamics systems, SIAM J. Math. Anal., 21 (1990), 593-630], including the interactions of strong shocks (shock reflections), vortex-vortex and shock-vortex etc. This study combines the theoretical results with the numerical simulations, and thus demonstrates what Ami Harten observed "for computational scientists there are two kinds of truth: the truth that you prove, and the truth you see when you compute" [33, Lax, Computational Fluid Dynamics, J. Sci. Comput., 31 (2007), 185-193].

Abstract. We determine completely the exact Riemann solutions for the system of Euler equations in a duct with discontinuous varying cross-section. The crucial point in solving the Riemann problem for hyperbolic system is the construction of the wave curves. To address the difficulty in the construction due to the nonstrict hyperbolicity of the underlying system, we introduce the L-M and R-M curves in the velocity-pressure phase plane. The behaviors of the L-M and R-M curves for six basic cases are fully analyzed. Furthermore, we observe that in certain cases the L-M and R-M curves contain the bifurcation which leads to the non-uniqueness of the Riemann solutions. Nevertheless, all possible Riemann solutions including classical as well as resonant solutions are solved in a uniform framework for any given initial data.

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We determine completely the Riemann solutions to the shallow water equations with a bottom step, including the dry bed problem. The nonstrict hyperbolicity of this first-order system of partial differential equations leads to resonant waves and nonunique solutions. To address these difficulties we construct the L-M and R-M curves in the state space. For the bottom step elevated from left to right, we classify the L-M curve into five different cases and the R-M curve into two different cases based on the subcritical and supercritical Froude number of the Riemann initial data as well as the jump of the bottom step. The behaviors of all basic cases of the L-M and R-M curves are fully analyzed. We observe that the non-uniqueness of the Riemann solutions is due to bifurcations on the L-M or R-M curves. The possible solutions including classical waves and resonant waves as well as dry bed state are solved in a uniform framework for any given Riemann initial data.

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