  In this paper we derive and analyse a class of linearly implicit schemes
which includes the one of Feistauer and Ku\v{c}era (JCP 2007) as well as the
class of RS-IMEX schemes. The implicit part is based on a Jacobian matrix which
is evaluated at a reference state. This state can be either the solution at the
old time level as in Feistauer and Ku\v{c}era (JCP 2007), or a numerical
approximation of the incompressible limit equations as in Zeifang et al.
(Commun. Comput. Phys. 2009), or possibly another state. Subsequently, it is
shown that this class of methods is asymptotically preserving under the
assumption of a discrete Hilbert expansion. For a one-dimensional setting with
some limitations on the reference state, the existence of a discrete Hilbert
expansion is shown.
