Accuracy of the Adaptive GRP Scheme and the Simulation of 2-D Riemann Problems for Compressible Euler Equations (Communications in Computational Physics, 2011)
The adaptive generalized Riemann problem (GRP) scheme for 2-D compressiblefluid 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 fora single shock, and the numerical instability of the almost stationary shock. In thispaper, we will provide the accuracy study and particularly show the performance insimulating 2-D complex wave configurations formulated with the 2-D Riemann problemsfor compressible Euler equations. For this purpose, we will first review the GRPscheme briefly when combined with the adaptive moving mesh technique and considerthe accuracy of the adaptive GRP scheme via the comparison with the explicitformulae of analytic solutions of planar rarefaction waves, planar shock waves, thecollapse problem of a wedge-shaped dam and the spiral formation problem. Then wesimulate the full set of wave configurations in the 2-D four-wave Riemann problemsfor compressible Euler equations [SIAM J. Math. Anal., 21 (1990), 593–630], includingthe interactions of strong shocks (shock reflections), vortex-vortex and shock-vortexetc. This study combines the theoretical results with the numerical simulations, andthus demonstrates what Ami Harten observed "for computational scientists there are twokinds of truth: the truth that you prove, and the truth you see when you compute" [J. Sci.Comput., 31 (2007), 185–193].