Commun. Comput. Phys., 10 (2011), pp. 577-606.

Accuracy of the Adaptive GRP Scheme and the Simulation of 2-D Riemann Problems for Compressible Euler Equations

Ee Han 1, Jiequan Li 2*, Huazhong Tang 3

1 IAN, Department of Mathematics, Magdeburg University, D-39106, Germany.
2 School of Mathematical Science, Beijing Normal University, Beijing 100875, China.
3 CAPT and LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China.

Received 28 April 2010; Accepted (in revised version) 30 July 2010
Available online 1 June 2011


The adaptive generalized Riemann problem (GRP) scheme for 2-D compressible fluid flows has been proposed in [J. Comput. Phys., 229 (2010), 1448-1466] 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 [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" [J. Sci. Comput., 31 (2007), 185-193].

AMS subject classifications: 65M06, 76M12, 35L60, 65M08

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Key words: Adaptive GRP scheme, 2-D Riemann problems, collapse of a wedge-shaped dam, spiral formation, shock reflections, vortex-shock interaction.

*Corresponding author.
Email: (E. Han), (J. Li), (H. Tang)

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