@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. }
}