  Roe scheme is known for its good performance in moderate-Mach-number flows.
However, this scheme and its extended versions suffers from many disastrous
problems, such as non-physical behavior, global cut-off, and checkerboard
problems, for incompressible flows; and shock instability, expansion shock, and
positively non-conservative problems for hypersonic flows. In this paper,
non-physical behavior problem, checkerboard problem, and main reason of shock
instability problem are due to that the Roe scheme cannot identify
multi-dimensional incompressible and compressible flows when normal Mach number
on the cell face tends to zero, and then leads to incorrect cross
modifications. Positively non-conservative problem is also identified as
another important reason for shock instability. Therefore, Mach number and an
assistant pressure-density-varying detector are introduced into the Roe scheme
to judge compressibility, positivity condition is satisfied by a simple
modification with minimal numerical dissipation increases and even with
possible decreases in numerical dissipation, the mechanism of the
preconditioned Roe scheme is introduced to suppress checkerboard problem, and
modified entropy fix and the rotated Riemann solver is combined with
complementary advantages as an assistant improvement for better robust. Based
on above improvements and previous developments for global cut-off and
expansion shock problems, an improvement Roe scheme for all Mach-number flow
(Roe-AM) is proposed to simultaneously overcome nearly all well-known drawbacks
of the classical Roe scheme. The Roe-AM scheme is simple, easy to implement,
computationally low-cost, robust, good extensibility, and free of empirical
parameters essentially, with increasing minimal numerical dissipation.
