Uppsala University
Department of Information Technology

### Technical Report 2002-029

# Domain Decomposition of the Padé Scheme and Pseudo-Spectral Method, Used in Vlasov Simulations

### Bengt Eliasson

October 2002

**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 Padé 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 Padé 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.

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