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IGPM510.pdf December 2020 
TITLE 
Asymptotic properties of a class of linearly implicit schemes for weakly compressible Euler equations 
AUTHORS 
Václav Kučera, Maria LukáčováMedvid‘ová, Sebastian Noelle, Jochen Schütz 
ABSTRACT 
In this paper we derive and analyse a class of linearly implicit
schemes which includes the one of Feistauer and Kuˇcera (JCP 2007) [9]
as well as the class of RSIMEX schemes [4, 17, 27, 28]. 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
[9], or a numerical approximation of the incompressible limit equations
as in [30], 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 onedimensional setting with some
limitations on the reference state, the existence of a discrete Hilbert
expansion is shown. 
KEYWORDS 
asymptotic preserving schemes, compressible Euler equations, LowMach limit Hilbert expansion 
DOI 
10.1007/s00211021012405 
PUBLICATION 
Numerische Mathematik, 150 (2022), 79–103. 