14 research outputs found
Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit
We investigate a projective integration scheme for a kinetic equation in the
limit of vanishing mean free path, in which the kinetic description approaches
a diffusion phenomenon. The scheme first takes a few small steps with a simple,
explicit method, such as a spatial centered flux/forward Euler time
integration, and subsequently projects the results forward in time over a large
time step on the diffusion time scale. We show that, with an appropriate choice
of the inner step size, the time-step restriction on the outer time step is
similar to the stability condition for the diffusion equation, whereas the
required number of inner steps does not depend on the mean free path. We also
provide a consistency result. The presented method is asymptotic-preserving, in
the sense that the method converges to a standard finite volume scheme for the
diffusion equation in the limit of vanishing mean free path. The analysis is
illustrated with numerical results, and we present an application to the
Su-Olson test
Projective integration for nonlinear BGK kinetic equations
Proceedings FVCA 8International audienceWe present a high-order, fully explicit, asymptotic-preserving projective integration scheme for the nonlinear BGK equation. The method first takes a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. Based on the spectrum of the linearized BGK operator, we deduce that, with an appropriate choice of inner step size, the time step restriction on the outer time step as well as the number of inner time steps is independent of the stiffness of the BGK source term. We illustrate the method with numerical results in one and two spatial dimensions
Asymptotic-Preserving Monte Carlo methods for transport equations in the diffusive limit
We develop a new Monte Carlo method that solves hyperbolic transport
equations with stiff terms, characterized by a (small) scaling parameter. In
particular, we focus on systems which lead to a reduced problem of parabolic
type in the limit when the scaling parameter tends to zero. Classical Monte
Carlo methods suffer of severe time step limitations in these situations, due
to the fact that the characteristic speeds go to infinity in the diffusion
limit. This makes the problem a real challenge, since the scaling parameter may
differ by several orders of magnitude in the domain. To circumvent these time
step limitations, we construct a new, asymptotic-preserving Monte Carlo method
that is stable independently of the scaling parameter and degenerates to a
standard probabilistic approach for solving the limiting equation in the
diffusion limit. The method uses an implicit time discretization to formulate a
modified equation in which the characteristic speeds do not grow indefinitely
when the scaling factor tends to zero. The resulting modified equation can
readily be discretized by a Monte Carlo scheme, in which the particles combine
a finite propagation speed with a time-step dependent diffusion term. We show
the performance of the method by comparing it with standard (deterministic)
approaches in the literature
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
On convergence of higher order schemes for the projective integration method for stiff ordinary differential equations
We present a convergence proof for higher order implementations of the
projective integration method (PI) for a class of deterministic multi-scale
systems in which fast variables quickly settle on a slow manifold. The error is
shown to contain contributions associated with the length of the microsolver,
the numerical accuracy of the macrosolver and the distance from the slow
manifold caused by the combined effect of micro- and macrosolvers,
respectively. We also provide stability conditions for the PI methods under
which the fast variables will not diverge from the slow manifold. We
corroborate our results by numerical simulations.Comment: 43 pages, 7 figures; accepted for publication in the Journal of
Computational and Applied Mathematic