17-03-2014, 10:18 PM
Abstract
In the paper, we present a new approach to solve the
Vlasov equation in two dimensions. First, we use the time
splitting method then we apply the ENO (essentially nonoscillatory)
scheme based on the idea of Harten to solve our
equation. We elaborate a transformation time-space to avoid
the mixed problem and reducing the error. This algorithm is
applied to the two dimensional Vlasov-Poisson system.
1. Introduction
In the last thirty years, various numerical methods have
been developed for solving the Vlasov-Poisson equations.
these methods can be parted in two main branches. the first
is the well-known particle-in-cell (PIC) method [1], [3]. This
one introduces the concept of pseudo-particles and relies on
the computation of their trajectories. This method has been
successfully used to simulate the behavior of collisionless
laboratory and space plasmas and provides accurate results
for modeling of large scale phenomena. Moreover suffer
from intrinsic drawbacks.
An alternative approach based on an Eulerian approach
consists in solving the Vlasov equation for the distribution
function [5], [8] .
There is a large number of competing finite difference
and related schemes for the solution of the Vlasov equation
suggested in the literature; we have chosen to use the
essentially nonoscillatory (“ENO”) schemes presented by
Osher and Shu [12] for the following reasons: First stable
schemes of arbitrarily high order accuracy exist, permitting
accurate solutions on coarse grids (which is critical to the
mesh refinement or coarsenment); Secondly versions exist
in any dimension so that we can extend our methodology to
the three-dimensional case straightforwardly.
In the paper, we present a new approach to solve the
Vlasov equation in two dimensions. First, we use the time
splitting method then we apply the ENO (essentially nonoscillatory)
scheme based on the idea of Harten to solve our
equation. We elaborate a transformation time-space to avoid
the mixed problem and reducing the error. This algorithm is
applied to the two dimensional Vlasov-Poisson system.
1. Introduction
In the last thirty years, various numerical methods have
been developed for solving the Vlasov-Poisson equations.
these methods can be parted in two main branches. the first
is the well-known particle-in-cell (PIC) method [1], [3]. This
one introduces the concept of pseudo-particles and relies on
the computation of their trajectories. This method has been
successfully used to simulate the behavior of collisionless
laboratory and space plasmas and provides accurate results
for modeling of large scale phenomena. Moreover suffer
from intrinsic drawbacks.
An alternative approach based on an Eulerian approach
consists in solving the Vlasov equation for the distribution
function [5], [8] .
There is a large number of competing finite difference
and related schemes for the solution of the Vlasov equation
suggested in the literature; we have chosen to use the
essentially nonoscillatory (“ENO”) schemes presented by
Osher and Shu [12] for the following reasons: First stable
schemes of arbitrarily high order accuracy exist, permitting
accurate solutions on coarse grids (which is critical to the
mesh refinement or coarsenment); Secondly versions exist
in any dimension so that we can extend our methodology to
the three-dimensional case straightforwardly.