| Abstract: | 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]. |
