Asymptotic properties of a class of linearly implicit schemes for weakly compressible Euler equations

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.