  There have been great efforts on the development of higher-order numerical
schemes for compressible Euler equations. The traditional tests mostly
targeting on the strong shock interactions alone may not be adequate to test
the performance of higher-order schemes. This study will introduce a few test
cases with a wide range of wave structures for testing higher-order schemes. As
reference solutions, all test cases will be calculated by our recently
developed two-stage fourth-order gas-kinetic scheme (GKS). All examples are
selected so that the numerical settings are very simple and any high order
accurate scheme can be straightly used for these test cases, and compare their
performance with the GKS solutions. The examples include highly oscillatory
solutions and the large density ratio problem in one dimensional case;
hurricane-like solutions, interactions of planar contact discontinuities (the
composite of entropy wave and vortex sheets) sheets with large Mach number
asymptotic, interaction of planar rarefaction waves with transition from
continuous flows to the presence of shocks, and other types of interactions of
two-dimensional planar waves. The numerical results from the fourth-order
gas-kinetic scheme provide reference solutions only. These benchmark test cases
will help CFD developers to validate and further develop their schemes to a
higher level of accuracy and robustness.
