Asymptotic properties of a class of linearly implicit schemes for weakly compressible Euler equations
Abstract
In this paper we derive and analyse a class of linearly implicit schemes which includes the one of Feistauer and Kucera (J Comput Phys 224:208-221, 2007) as well as the class of RS-IMEX schemes (Schutz and Noelle in J Sci Comp 64:522-540, 2015; Kaiser et al. in J Sci Comput 70:1390-1407, 2017; Bispen et al. in Commun Comput Phys 16:307-347, 2014; Zakerzadeh in ESAIM Math Model Numer Anal 53:893-924, 2019). 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 Kucera (2007), or a numerical approximation of the incompressible limit equations as in Zeifang et al. (Commun Comput Phys 27:292-320, 2020), 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.