53 research outputs found
An implicit-explicit solver for a two-fluid single-temperature model
We present an implicit-explicit finite volume scheme for two-fluid
single-temperature flow in all Mach number regimes which is based on a
symmetric hyperbolic thermodynamically compatible description of the fluid
flow. The scheme is stable for large time steps controlled by the interface
transport and is computational efficient due to a linear implicit character.
The latter is achieved by linearizing along constant reference states given by
the asymptotic analysis of the single-temperature model. Thus, the use of a
stiffly accurate IMEX Runge Kutta time integration and the centered treatment
of pressure based quantities provably guarantee the asymptotic preserving
property of the scheme for weakly compressible Euler equations with variable
volume fraction. The properties of the first and second order scheme are
validated by several numerical test cases
Flux Splitting for stiff equations: A notion on stability
For low Mach number flows, there is a strong recent interest in the
development and analysis of IMEX (implicit/explicit) schemes, which rely on a
splitting of the convective flux into stiff and nonstiff parts. A key
ingredient of the analysis is the so-called Asymptotic Preserving (AP)
property, which guarantees uniform consistency and stability as the Mach number
goes to zero. While many authors have focussed on asymptotic consistency, we
study asymptotic stability in this paper: does an IMEX scheme allow for a CFL
number which is independent of the Mach number? We derive a stability criterion
for a general linear hyperbolic system. In the decisive eigenvalue analysis,
the advective term, the upwind diffusion and a quadratic term stemming from the
truncation in time all interact in a subtle way. As an application, we show
that a new class of splittings based on characteristic decomposition, for which
the commutator vanishes, avoids the deterioration of the time step which has
sometimes been observed in the literature
High Order Asymptotic Preserving and Classical Semi-implicit RK Schemes for the Euler-Poisson System in the Quasineutral Limit
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
An all speed second order well-balanced IMEX relaxation scheme for the Euler equations with gravity
We present an implicit-explicit well-balanced finite volume scheme for the Euler equations with a gravitational source term which is able to deal also with low Mach flows. To visualize the different scales we use the non-dimensionalized equations on which we apply a pressure splitting and a Suliciu relaxation. On the resulting model, we apply a splitting of the flux into a linear implicit and an non-linear explicit part that leads to a scale independent time-step. The explicit step consists of a Godunov type method based on an approximative Riemann solver where the source term is included in the flux formulation. We develop the method for a first order scheme and give an extension to second order. Both schemes are designed to be well-balanced, preserve the positivity of density and internal energy and have a scale independent diffusion. We give the low Mach limit equations for well-prepared data and show that the scheme is asymptotic preserving. These properties are numerically validated by various test cases
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