2,260 research outputs found
Numerical schemes for kinetic equations in the diffusion and anomalous diffusion limits. Part I: the case of heavy-tailed equilibrium
In this work, we propose some numerical schemes for linear kinetic equations
in the diffusion and anomalous diffusion limit. When the equilibrium
distribution function is a Maxwellian distribution, it is well known that for
an appropriate time scale, the small mean free path limit gives rise to a
diffusion type equation. However, when a heavy-tailed distribution is
considered, another time scale is required and the small mean free path limit
leads to a fractional anomalous diffusion equation. Our aim is to develop
numerical schemes for the original kinetic model which works for the different
regimes, without being restricted by stability conditions of standard explicit
time integrators. First, we propose some numerical schemes for the diffusion
asymptotics; then, their extension to the anomalous diffusion limit is studied.
In this case, it is crucial to capture the effect of the large velocities of
the heavy-tailed equilibrium, so that some important transformations of the
schemes derived for the diffusion asymptotics are needed. As a result, we
obtain numerical schemes which enjoy the Asymptotic Preserving property in the
anomalous diffusion limit, that is: they do not suffer from the restriction on
the time step and they degenerate towards the fractional diffusion limit when
the mean free path goes to zero. We also numerically investigate the uniform
accuracy and construct a class of numerical schemes satisfying this property.
Finally, the efficiency of the different numerical schemes is shown through
numerical experiments
Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit
We construct numerical schemes to solve kinetic equations with anomalous
diffusion scaling. When the equilibrium is heavy-tailed or when the collision
frequency degenerates for small velocities, an appropriate scaling should be
made and the limit model is the so-called anomalous or fractional diffusion
model. Our first scheme is based on a suitable micro-macro decomposition of the
distribution function whereas our second scheme relies on a Duhamel formulation
of the kinetic equation. Both are \emph{Asymptotic Preserving} (AP): they are
consistent with the kinetic equation for all fixed value of the scaling
parameter and degenerate into a consistent scheme solving the
asymptotic model when tends to . The second scheme enjoys the
stronger property of being uniformly accurate (UA) with respect to
. The usual AP schemes known for the classical diffusion limit
cannot be directly applied to the context of anomalous diffusion scaling, since
they are not able to capture the important effects of large and small
velocities. We present numerical tests to highlight the efficiency of our
schemes
Fractional statistical dynamics and fractional kinetics
We apply the subordination principle to construct kinetic fractional
statistical dynamics in the continuum in terms of solutions to Vlasov-type
hierarchies. As a by-product we obtain the evolution of the density of
particles in the fractional kinetics in terms of a non-linear Vlasov-type
kinetic equation. As an application we study the intermittency of the
fractional mesoscopic dynamics.Comment: Published in Methods of Functional Analysis and Topology (MFAT),
available at http://mfat.imath.kiev.ua/article/?id=890. arXiv admin note:
text overlap with arXiv:1604.0380
High Order Asymptotic Preserving DG-IMEX Schemes for Discrete-Velocity Kinetic Equations in a Diffusive Scaling
In this paper, we develop a family of high order asymptotic preserving
schemes for some discrete-velocity kinetic equations under a diffusive scaling,
that in the asymptotic limit lead to macroscopic models such as the heat
equation, the porous media equation, the advection-diffusion equation, and the
viscous Burgers equation. Our approach is based on the micro-macro
reformulation of the kinetic equation which involves a natural decomposition of
the equation to the equilibrium and non-equilibrium parts. To achieve high
order accuracy and uniform stability as well as to capture the correct
asymptotic limit, two new ingredients are employed in the proposed methods:
discontinuous Galerkin spatial discretization of arbitrary order of accuracy
with suitable numerical fluxes; high order globally stiffly accurate
implicit-explicit Runge-Kutta scheme in time equipped with a properly chosen
implicit-explicit strategy. Formal asymptotic analysis shows that the proposed
scheme in the limit of epsilon -> 0 is an explicit, consistent and high order
discretization for the limiting equation. Numerical results are presented to
demonstrate the stability and high order accuracy of the proposed schemes
together with their performance in the limit
Numerical schemes for kinetic equations in the anomalous diffusion limit. Part II: degenerate collision frequency.
International audienceIn this work, which is the continuation of [9], we propose numerical schemes for linear kinetic equation which are able to deal with the fractional diffusion limit. When the collision frequency degenerates for small velocities it is known that for an appropriate time scale, the small mean free path limit leads to an anomalous diffusion equation. From a numerical point of view, this degeneracy gives rise to an additional stiffness that must be treated in a suitable way to avoid a prohibitive computational cost. Our aim is therefore to construct a class of numerical schemes which are able to undertake these stiffness. This means that the numerical schemes are able to capture the effect of small velocities in the small mean free path limit with a fixed set of numerical parameters. Various numerical tests are performed to illustrate the efficiency of our methods in this context
Numerical schemes for kinetic equation with diffusion limit and anomalous time scale
In this work, we propose numerical schemes for linear kinetic equation , which are able to deal with a diffusion limit and an anomalous time scale of the form ε 2 (1 + |ln(ε)|). When the equilibrium distribution function is a heavy-tailed function, it is known that for an appropriate time scale, the mean-free-path limit leads either to diffusion or fractional diffusion equation, depending on the tail of the equilibrium. The bifurcation between these two limits is the classical diffusion limit with anomalous time scale treated in this work. Our aim is to develop numerical schemes which work for the different regimes, with no restriction on the numerical parameters. Indeed, the degen-eracy ε → 0 makes the kinetic equation stiff. From a numerical point of view, it is necessary to construct schemes able to undertake this stiffness to avoid the increase of computational cost. In this case, it is crucial to capture numerically the effects of the large velocities of the heavy-tailed equilibrium. Since the degeneracy towards the diffusion limit is very slow, it is also essential to respect the asymptotic behavior of the solution, and not only the limit. Various numerical tests are performed to illustrate the efficiency of our methods in this context
An asymptotic preserving scheme for kinetic models with singular limit
We propose a new class of asymptotic preserving schemes to solve kinetic
equations with mono-kinetic singular limit. The main idea to deal with the
singularity is to transform the equations by appropriate scalings in velocity.
In particular, we study two biologically related kinetic systems. We derive the
scaling factors and prove that the rescaled solution does not have a singular
limit, under appropriate spatial non-oscillatory assumptions, which can be
verified numerically by a newly developed asymptotic preserving scheme. We set
up a few numerical experiments to demonstrate the accuracy, stability,
efficiency and asymptotic preserving property of the schemes.Comment: 24 pages, 6 figure
- …