6,230 research outputs found
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
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
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