139 research outputs found
An asymptotically stable scheme for diffusive coagulation-fragmentation models
This paper is devoted to the analysis of a numerical scheme for the
coagulation and fragmentation equation with diffusion in space. A finite volume
scheme is developed, based on a conservative formulation of the space
nonhomogeneous coagulation-fragmentation model, it is shown that the scheme
preserves positivity, total volume and global steady states. Finally, several
numerical simulations are performed to investigate the long time behavior of
the solution
On deterministic approximation of the Boltzmann equation in a bounded domain
In this paper we present a fully deterministic method for the numerical
solution to the Boltzmann equation of rarefied gas dynamics in a bounded domain
for multi-scale problems. Periodic, specular reflection and diffusive boundary
conditions are discussed and investigated numerically. The collision operator
is treated by a Fourier approximation of the collision integral, which
guarantees spectral accuracy in velocity with a computational cost of
, where is the number of degree of freedom in velocity space.
This algorithm is coupled with a second order finite volume scheme in space and
a time discretization allowing to deal for rarefied regimes as well as their
hydrodynamic limit. Finally, several numerical tests illustrate the efficiency
and accuracy of the method for unsteady flows (Poiseuille flows, ghost effects,
trend to equilibrium)
An Asymptotic Preserving Scheme for the ES-BGK model
In this paper, we study a time discrete scheme for the initial value problem
of the ES-BGK kinetic equation. Numerically solving these equations are
challenging due to the nonlinear stiff collision (source) terms induced by
small mean free or relaxation time. We study an implicit-explicit (IMEX) time
discretization in which the convection is explicit while the relaxation term is
implicit to overcome the stiffness. We first show how the implicit relaxation
can be solved explicitly, and then prove asymptotically that this time
discretization drives the density distribution toward the local Maxwellian when
the mean free time goes to zero while the numerical time step is held fixed.
This naturally imposes an asymptotic-preserving scheme in the Euler limit. The
scheme so designed does not need any nonlinear iterative solver for the
implicit relaxation term. Moreover, it can capture the macroscopic fluid
dynamic (Euler) limit even if the small scale determined by the Knudsen number
is not numerically resolved. We also show that it is consistent to the
compressible Navier-Stokes equations if the viscosity and heat conductivity are
numerically resolved. Several numerical examples, in both one and two space
dimensions, are used to demonstrate the desired behavior of this scheme
Mixed semi-Lagrangian/finite difference methods for plasma simulations
In this paper, we present an efficient algorithm for the long time behavior
of plasma simulations. We will focus on 4D drift-kinetic model, where the
plasma's motion occurs in the plane perpendicular to the magnetic field and can
be governed by the 2D guiding-center model.
Hermite WENO reconstructions, already proposed in \cite{YF15}, are applied
for solving the Vlasov equation. Here we consider an arbitrary computational
domain with an appropriate numerical method for the treatment of boundary
conditions.
Then we apply this algorithm for plasma turbulence simulations. We first
solve the 2D guiding-center model in a D-shape domain and investigate the
numerical stability of the steady state. Then, the 4D drift-kinetic model is
studied with a mixed method, i.e. the semi-Lagrangian method in linear phase
and finite difference method during the nonlinear phase. Numerical results show
that the mixed method is efficient and accurate in linear phase and it is much
stable during the nonlinear phase. Moreover, in practice it has better
conservation properties.Comment: arXiv admin note: text overlap with arXiv:1312.448
A Hierarchy of Hybrid Numerical Methods for Multi-Scale Kinetic Equations
In this paper, we construct a hierarchy of hybrid numerical methods for
multi-scale kinetic equations based on moment realizability matrices, a concept
introduced by Levermore, Morokoff and Nadiga. Following such a criterion, one
can consider hybrid scheme where the hydrodynamic part is given either by the
compressible Euler or Navier-Stokes equations, or even with more general
models, such as the Burnett or super-Burnett systems.Comment: 27 pages, edit: typo and metadata chang
Conservative and non-conservative methods based on hermite weighted essentially-non-oscillatory reconstruction for Vlasov equations
We introduce a WENO reconstruction based on Hermite interpolation both for
semi-Lagrangian and finite difference methods. This WENO reconstruction
technique allows to control spurious oscillations. We develop third and fifth
order methods and apply them to non-conservative semi-Lagrangian schemes and
conservative finite difference methods. Our numerical results will be compared
to the usual semi-Lagrangian method with cubic spline reconstruction and the
classical fifth order WENO finite difference scheme. These reconstructions are
observed to be less dissipative than the usual weighted essentially non-
oscillatory procedure. We apply these methods to transport equations in the
context of plasma physics and the numerical simulation of turbulence phenomena
Numerical study of a nonlinear heat equation for plasma physics
This paper is devoted to the numerical approximation of a nonlinear
temperature balance equation, which describes the heat evolution of a
magnetically confined plasma in the edge region of a tokamak. The nonlinearity
implies some numerical difficulties, in particular long time behavior, when
solved with standard methods. An efficient numerical scheme is presented in
this paper, based on a combination of a directional splitting scheme and the
IMEX scheme introduced in [Filbet and Jin
On steady-state preserving spectral methods for homogeneous Boltzmann equations
In this note, we present a general way to construct spectral methods for the
collision operator of the Boltzmann equation which preserves exactly the
Maxwellian steady-state of the system. We show that the resulting method is
able to approximate with spectral accuracy the solution uniformly in time.Comment: 7 pages, 3 figure
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