1,196 research outputs found
High order asymptotic-preserving schemes for the Boltzmann equation
In this note we discuss the construction of high order asymptotic preserving
numerical schemes for the Boltzmann equation. The methods are based on the use
of Implicit-Explicit (IMEX) Runge-Kutta methods combined with a penalization
technique recently introduced in [F. Filbet, S. Jin: A class of asymptotic
preserving schemes for kinetic equations and related problems with stiff
sources,J. Comp. Phys. 229, (2010), pp. 7625-7648.]
A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources
In this paper, we propose a general framework to design asymptotic preserving
schemes for the Boltzmann kinetic kinetic and related equations. Numerically
solving these equations are challenging due to the nonlinear stiff collision
(source) terms induced by small mean free or relaxation time. We propose to
penalize the nonlinear collision term by a BGK-type relaxation term, which can
be solved explicitly even if discretized implicitly in time. Moreover, the
BGK-type relaxation operator helps to drive the density distribution toward the
local Maxwellian, thus natually imposes an asymptotic-preserving scheme in the
Euler limit. The scheme so designed does not need any nonlinear iterative
solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly
small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler)
limit even if the small scale determined by the Knudsen number is not
numerically resolved. It is also consistent to the compressible Navier-Stokes
equations if the viscosity and heat conductivity are numerically resolved. The
method is applicable to many other related problems, such as hyperbolic systems
with stiff relaxation, and high order parabilic equations
Exponential Runge-Kutta methods for stiff kinetic equations
We introduce a class of exponential Runge-Kutta integration methods for
kinetic equations. The methods are based on a decomposition of the collision
operator into an equilibrium and a non equilibrium part and are exact for
relaxation operators of BGK type. For Boltzmann type kinetic equations they
work uniformly for a wide range of relaxation times and avoid the solution of
nonlinear systems of equations even in stiff regimes. We give sufficient
conditions in order that such methods are unconditionally asymptotically stable
and asymptotic preserving. Such stability properties are essential to guarantee
the correct asymptotic behavior for small relaxation times. The methods also
offer favorable properties such as nonnegativity of the solution and entropy
inequality. For this reason, as we will show, the methods are suitable both for
deterministic as well as probabilistic numerical techniques
A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation
In this paper we consider the development of Implicit-Explicit (IMEX)
Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such
systems the scaling depends on an additional parameter which modifies the
nature of the asymptotic behavior which can be either hyperbolic or parabolic.
Because of the multiple scalings, standard IMEX Runge-Kutta methods for
hyperbolic systems with relaxation loose their efficiency and a different
approach should be adopted to guarantee asymptotic preservation in stiff
regimes. We show that the proposed approach is capable to capture the correct
asymptotic limit of the system independently of the scaling used. Several
numerical examples confirm our theoretical analysis
Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit
We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic
systems with stiff relaxation in the so-called diffusion limit. In such regime
the system relaxes towards a convection-diffusion equation. The first objective
of the paper is to show that traditional partitioned IMEX R-K schemes will
relax to an explicit scheme for the limit equation with no need of modification
of the original system. Of course the explicit scheme obtained in the limit
suffers from the classical parabolic stability restriction on the time step.
The main goal of the paper is to present an approach, based on IMEX R-K
schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the
convection-diffusion equation, in which the diffusion is treated implicitly.
This is achieved by an original reformulation of the problem, and subsequent
application of IMEX R-K schemes to it. An analysis on such schemes to the
reformulated problem shows that the schemes reduce to IMEX R-K schemes for the
limit equation, under the same conditions derived for hyperbolic relaxation.
Several numerical examples including neutron transport equations confirm the
theoretical analysis
Asymptotic preserving Implicit-Explicit Runge-Kutta methods for non linear kinetic equations
We discuss Implicit-Explicit (IMEX) Runge Kutta methods which are
particularly adapted to stiff kinetic equations of Boltzmann type. We consider
both the case of easy invertible collision operators and the challenging case
of Boltzmann collision operators. We give sufficient conditions in order that
such methods are asymptotic preserving and asymptotically accurate. Their
monotonicity properties are also studied. In the case of the Boltzmann
operator, the methods are based on the introduction of a penalization technique
for the collision integral. This reformulation of the collision operator
permits to construct penalized IMEX schemes which work uniformly for a wide
range of relaxation times avoiding the expensive implicit resolution of the
collision operator. Finally we show some numerical results which confirm the
theoretical analysis
Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation
We consider the development of high order space and time numerical methods
based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic
systems with relaxation. More specifically, we consider hyperbolic balance laws
in which the convection and the source term may have very different time and
space scales. As a consequence the nature of the asymptotic limit changes
completely, passing from a hyperbolic to a parabolic system. From the
computational point of view, standard numerical methods designed for the
fluid-dynamic scaling of hyperbolic systems with relaxation present several
drawbacks and typically lose efficiency in describing the parabolic limit
regime. In this work, in the context of Implicit-Explicit linear multistep
methods we construct high order space-time discretizations which are able to
handle all the different scales and to capture the correct asymptotic behavior,
independently from its nature, without time step restrictions imposed by the
fast scales. Several numerical examples confirm the theoretical analysis
Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy
In this paper we develop high-order asymptotic-preserving methods for the
spatially inhomogeneous quantum Boltzmann equation. We follow the work in Li
and Pareschi, where asymptotic preserving exponential Runge-Kutta methods for
the classical inhomogeneous Boltzmann equation were constructed. A major
difficulty here is related to the non Gaussian steady states characterizing the
quantum kinetic behavior. We show that the proposed schemes work with
high-order accuracy uniformly in time for all Planck constants ranging from
classical regime to quantum regime, and all Knudsen numbers ranging from
kinetic regime to fluid regime. Computational results are presented for both
Bose gas and Fermi gas
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