  In this paper, we propose a new high order semi-implicit scheme for the all
Mach full Euler equations of gas dynamics. Material waves are treated
explicitly, while acoustic waves are treated implicitly, thus avoiding severe
CFL restrictions for low Mach flows. High order accuracy in time is obtained by
semi-implicit temporal integrator based on the IMEX Runge-Kutta (IMEX-RK)
framework. High order in space is achieved by finite difference WENO schemes
with characteristic-wise reconstructions adapted to the semi-implicit IMEX-RK
time discretization. Type A IMEX schemes are constructed to handle not
well-prepared initial conditions. Besides, these schemes are proven to be
asymptotic preserving and asymptotically accurate as the Mach number vanishes
for well-prepared initial conditions. Divergence-free property of the
time-discrete schemes is proved. The proposed scheme can also well capture
discontinuous solutions in the compressible regime, especially for two
dimensional Riemann problems. Numerical tests in one and two space dimensions
will illustrate the effectiveness of the proposed schemes.
