17 research outputs found
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
All speed scheme for the low mach number limit of the Isentropic Euler equation
An all speed scheme for the Isentropic Euler equation is presented in this
paper. When the Mach number tends to zero, the compressible Euler equation
converges to its incompressible counterpart, in which the density becomes a
constant. Increasing approximation errors and severe stability constraints are
the main difficulty in the low Mach regime. The key idea of our all speed
scheme is the special semi-implicit time discretization, in which the low Mach
number stiff term is divided into two parts, one being treated explicitly and
the other one implicitly. Moreover, the flux of the density equation is also
treated implicitly and an elliptic type equation is derived to obtain the
density. In this way, the correct limit can be captured without requesting the
mesh size and time step to be smaller than the Mach number. Compared with
previous semi-implicit methods, nonphysical oscillations can be suppressed. We
develop this semi-implicit time discretization in the framework of a first
order local Lax-Friedrich (LLF) scheme and numerical tests are displayed to
demonstrate its performances
Degenerate anisotropic elliptic problems and magnetized plasma simulations
This paper is devoted to the numerical approximation of a degenerate
anisotropic elliptic problem. The numerical method is designed for arbitrary
space-dependent anisotropy directions and does not require any specially
adapted coordinate system. It is also designed to be equally accurate in the
strongly and the mildly anisotropic cases. The method is applied to the
Euler-Lorentz system, in the drift-fluid limit. This system provides a model
for magnetized plasmas
An asymptotic preserving scheme for strongly anisotropic elliptic problems
In this article we introduce an asymptotic preserving scheme designed to
compute the solution of a two dimensional elliptic equation presenting large
anisotropies. We focus on an anisotropy aligned with one direction, the
dominant part of the elliptic operator being supplemented with Neumann boundary
conditions. A new scheme is introduced which allows an accurate resolution of
this elliptic equation for an arbitrary anisotropy ratio.Comment: 21 page
Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit
This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB)
model of plasma physics. This model consists of the pressureless gas dynamics
equations coupled with the Poisson equation and where the Boltzmann relation
relates the potential to the electron density. If the quasi-neutral assumption
is made, the Poisson equation is replaced by the constraint of zero local
charge and the model reduces to the Isothermal Compressible Euler (ICE) model.
We compare a numerical strategy based on the EPB model to a strategy using a
reformulation (called REPB formulation). The REPB scheme captures the
quasi-neutral limit more accurately
Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations
The present paper introduces an efficient and accurate numerical scheme for
the solution of a highly anisotropic elliptic equation, the anisotropy
direction being given by a variable vector field. This scheme is based on an
asymptotic preserving reformulation of the original system, permitting an
accurate resolution independently of the anisotropy strength and without the
need of a mesh adapted to this anisotropy. The counterpart of this original
procedure is the larger system size, enlarged by adding auxiliary variables and
Lagrange multipliers. This Asymptotic-Preserving method generalizes the method
investigated in a previous paper [arXiv:0903.4984v2] to the case of an
arbitrary anisotropy direction field
Numerical resolution of an anisotropic non-linear diffusion problem
International audienceThis paper is devoted to the numerical resolution of an anisotropic non-linear diffusion problem involving a small parameter ε, defined as the anisotropy strength reciprocal. In this work, the anisotropy is carried by a variable vector function b. The equation being supplemented with Neumann boundary conditions, the limit ε → 0 is demonstrated to be a singular perturbation of the original diffusion equation. To address efficiently this problem, an Asymptotic-Preserving scheme is derived. This numerical method does not require the use of coordinates adapted to the anisotropy direction and exhibits an accuracy as well as a computational cost independent of the anisotropy strength