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Roe scheme is known for its good performance in moderate-Mach-number flows.However, this scheme and its extended versions suffers from many disastrousproblems, such as non-physical behavior, global cut-off, and checkerboardproblems, for incompressible flows; and shock instability, expansion shock, andpositively non-conservative problems for hypersonic flows. In this paper,non-physical behavior problem, checkerboard problem, and main reason of shockinstability problem are due to that the Roe scheme cannot identifymulti-dimensional incompressible and compressible flows when normal Mach numberon the cell face tends to zero, and then leads to incorrect crossmodifications. Positively non-conservative problem is also identified asanother important reason for shock instability. Therefore, Mach number and anassistant pressure-density-varying detector are introduced into the Roe schemeto judge compressibility, positivity condition is satisfied by a simplemodification with minimal numerical dissipation increases and even withpossible decreases in numerical dissipation, the mechanism of thepreconditioned Roe scheme is introduced to suppress checkerboard problem, andmodified entropy fix and the rotated Riemann solver is combined withcomplementary advantages as an assistant improvement for better robust. Basedon above improvements and previous developments for global cut-off andexpansion shock problems, an improvement Roe scheme for all Mach-number flow(Roe-AM) is proposed to simultaneously overcome nearly all well-known drawbacksof the classical Roe scheme. The Roe-AM scheme is simple, easy to implement,computationally low-cost, robust, good extensibility, and free of empiricalparameters essentially, with increasing minimal numerical dissipation.

An Improved Roe Scheme for All Mach-Number Flows Simultaneously Curing Known Problems