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 Kučera (J Comput Phys 224:208–221, 2007) as well as the class of RS-IMEX schemes (Schütz 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 Kučera (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.

Low-Mach consistency of a class of linearly implicit schemes for the compressible Euler equations

“…Several details of the analysis were left out due to limited space, including the proof of Theorem 4. These are contained in the authors' paper [6]. One important topic that was left out of this brief overview was the question of existence of the Hilbert expansion with respect to ε of the form (12).…”

“…Due to lack of space, the analysis cannot be performed in its entirety, cf. [6] for details. The purpose here is to demonstrate the basic principles of performing such an analysis.…”

“…similarly, this is assumed for the reference state w R . The existence of the Hilbert expansion is proved under simplifying assumptions in [6].…”

An all Mach number finite volume method for isentropic two-phase flow

“…The numerical results are displayed in Figure 5. Figure 5a presents a zoom on the acoustic waves of phase one in the domain [23,27] whereas the acoustic waves of phase two have already reached the domains [3,12] and [37,49] as depicted in Figure 5b. We run the simulation with three different time steps.…”

“…The profound knowledge of the structure of well-prepared initial data can be used to construct an AP scheme by applying a reference solution (RS)-IMEX approach. This approach was successfully applied to construct AP schemes for the (isentropic) Euler equations [5,22,27,40]. Here, we only linearise the nonlinear pressure based terms around a reference state given by well-prepared data.…”