@TechReport{ it:2002-029,
author = {Bengt Eliasson},
title = {Domain Decomposition of the {P}ad{\'e} Scheme and
Pseudo-Spectral Method, Used in {V}lasov Simulations},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2002},
number = {2002-029},
month = oct,
abstract = {In order to evaluate parallel algorithms for solving the
Vlasov equation numerically in multiple dimensions, the
algorithm for solving the one-dimensional Vlasov equation
numerically has been parallelised. The one-dimensional
Vlasov equation leads to a problem in the two-dimensional
phase space $(x,v)$, plus time. The parallelisation is
performed by domain decomposition to a rectangular
processor grid. Derivatives in $x$ space are calculated by
a pseudo-spectral method, where FFTs are used to perform
discrete Fourier transforms. In velocity $v$ space a
Fourier method is used, together with the compact Pad{\'e}
scheme for calculating derivatives, leading to a large
number of tri-diagonal linear systems to be solved. The
parallelisation of the tri-diagonal systems in the Fourier
transformed velocity space can be performed efficiently by
the method of domain decomposition. The domain
decomposition gives rise to Schur complement systems, which
are tri-diagonal, symmetric and strongly diagonally
dominant, making it possible to solve these systems with a
few Jacobi iterations. Therefore, the parallel efficiency
of the semi-implicit Pad{\'e} scheme is comparable to the
parallel efficiency of explicit difference schemes. The
parallelisation in $x$ space is less effective due to the
FFTs used. The code has been tested on shared memory
computers, on clusters of computers, and with the help of
the Globus toolkit for communication over the Internet. }
}