  In this paper, the design and analysis of high order accurate IMEX finite
volume schemes for the compressible Euler-Poisson (EP) equations in the
quasineutral limit is presented. As the quasineutral limit is singular for the
governing equations, the time discretisation is tantamount to achieving an
accurate numerical method. To this end, the EP system is viewed as a
differential algebraic equation system (DAEs) via the method of lines. As a
consequence of this vantage point, high order linearly semi-implicit (SI) time
discretisation are realised by employing a novel combination of the direct
approach used for implicit discretisation of DAEs and, two different classes of
IMEX-RK schemes: the additive and the multiplicative. For both the time
discretisation strategies, in order to account for rapid plasma oscillations in
quasineutral regimes, the nonlinear Euler fluxes are split into two different
combinations of stiff and non-stiff components. The high order scheme resulting
from the additive approach is designated as a classical scheme while the one
generated by the multiplicative approach possesses the asymptotic preserving
(AP) property. Time discretisations for the classical and the AP schemes are
performed by standard IMEX-RK and SI-IMEX-RK methods, respectively so that the
stiff terms are treated implicitly and the non-stiff ones explicitly. In order
to discretise in space a Rusanov-type central flux is used for the non-stiff
part, and simple central differencing for the stiff part. AP property is also
established for the space-time fully-discrete scheme obtained using the
multiplicative approach. Results of numerical experiments are presented, which
confirm that the high order schemes based on the SI-IMEX-RK time discretisation
achieve uniform second order convergence with respect to the Debye length and
are AP in the quasineutral limit.
