  In recent years, Roe-type schemes based on different ideas have been
developed for all-speed flows, such as the preconditioned Roe, the All-Speed
Roe, Thornber's modified Roe and the LM-Roe schemes. This work explores why
these schemes succeed or fail with the accuracy and checkerboard problems.
Comparison and analysis show that the accuracy and checkerboard problems are
caused by the order of the sound speed being too large and too small in the
coefficients of the velocity-derivative and pressure-derivative dissipation
terms, respectively. These problems can be resolved by choosing coefficients
with zero-order sound speed. In addition, to avoid the negative effects of the
global cut-off strategy on accuracy while maintaining computational stability,
the sound speed terms in the numerator of the coefficients can be determined by
local variables, while those in the denominator remain the global cut-off. Two
novel schemes are proposed as examples to demonstrate how these ideas can be
applied to construct more satisfactory schemes for all-speed flows. Asymptotic
analysis and numerical experiments support the theoretical analysis and the
rules obtained in the work.
