Degond, P; Jin, S; Liu, JG, *Mach-number uniform asymptotic- preserving Gauge schemes for compressible flows*,
Bulletin of the Institute of Mathematics Academia Sinica (New Series), vol. 2
(2007),
pp. 851-892 .
**Abstract:**

*We present novel algorithms for compressible flows that are
efficient for all Mach numbers. The approach is based on several
ingredients: semi-implicit schemes, the gauge decomposition of the
velocity field and a second order formulation of the density
equation (in the isentropic case) and of the energy equation (in the
full Navier-Stokes case). Additionally, we show that our approach
corresponds to a micro-macro decomposition of the model, where the
macro field corresponds to the incompressible component satisfying a
perturbed low Mach number limit equation and the micro field is the
potential component of the velocity. Finally, we also use the
conservative variables in order to obtain a proper conservative
formulation of the equations when the Mach number is order unity. We
successively consider the isentropic case, the full Navier-Stokes
case, and the isentropic Navier-Stokes-Poisson case. In this work,
we only concentrate on the question of the time discretization and
show that the proposed method leads to Asymptotic Preserving
schemes for compressible flows in the low Mach number limit.*