98 research outputs found
Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations
In multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations
Long term analysis of splitting methods for charged-particle dynamics
In this paper, we rigorously analyze the energy, momentum and magnetic moment
behaviours of two splitting methods for solving charged-particle dynamics. The
near-conservations of these invariants are given for the system under constant
magnetic field or quadratic electric potential. By the approach named as
backward error analysis, we derive the modified equations and modified
invariants of the splitting methods and based on which, the near-conservations
over long times are proved. Some numerical experiments are presented to
demonstrate these long time behaviours
An Asymptotic Preserving Scheme for the Euler equations in a strong magnetic field
This paper is concerned with the numerical approximation of the isothermal
Euler equations for charged particles subject to the Lorentz force. When the
magnetic field is large, the so-called drift-fluid approximation is obtained.
In this limit, the parallel motion relative to the magnetic field direction
splits from perpendicular motion and is given implicitly by the constraint of
zero total force along the magnetic field lines. In this paper, we provide a
well-posed elliptic equation for the parallel velocity which in turn allows us
to construct an Asymptotic-Preserving (AP) scheme for the Euler-Lorentz system.
This scheme gives rise to both a consistent approximation of the Euler-Lorentz
model when epsilon is finite and a consistent approximation of the drift limit
when epsilon tends to 0. Above all, it does not require any constraint on the
space and time steps related to the small value of epsilon. Numerical results
are presented, which confirm the AP character of the scheme and its Asymptotic
Stability
Non-thermal particle acceleration and power-law tails via relaxation to universal Lynden-Bell equilibria
Collisionless and weakly collisional plasmas often exhibit non-thermal
quasi-equilibria. Among these quasi-equilibria, distributions with power-law
tails are ubiquitous. It is shown that the statistical-mechanical approach
originally suggested by Lynden-Bell (1967) can easily recover such power-law
tails. Moreover, we show that, despite the apparent diversity of Lynden-Bell
equilibria, a generic form of the equilibrium distribution at high energies is
a `hard' power-law tail , where is the
particle energy. The shape of the `core' of the distribution, located at low
energies, retains some dependence on the initial condition but it is the tail
(or `halo') that contains most of the energy. Thus, a degree of universality
exists in collisionless plasmas.Comment: 33 pages, 5 figure
Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence
We consider the closure problem for turbulence in the dry convective atmospheric boundary
layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large
plumes in the well mixed middle part up to the inversion that separates the CBL from the
stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF
approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02)
that additionally includes a term for background turbulence. Thus an exact solution is derived
and all higher order moments (HOMs) are explained by second order moments, correlation
coefficients and the skewness. The solution provides a proof of the extended universality
hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi-
normality of FOM). This refined hypothesis states that CBL turbulence can be considered as
result of a linear interpolation between the Gaussian and the very skewed turbulence regimes.
Although the extended universality hypothesis was confirmed by results of field
measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained
unexplained. These are now answered by the new model including the reasons of the
universality of the functional form of the HOMs, the significant scatter of the values of the
coefficients and the source of the magic of the linear interpolation. Finally, the closures
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predicted by the model are tested against measurements and LES data. Some of the other
issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area
coverage parameters of plumes (so called filling factors) with HOM will be discussed also
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