A RIGOROUS ANALYSIS OF TIME DOMAIN PARALLELISM∗

A. DESHPANDE, S. MALHOTRA, M. H. SCHULTZ, C. C. DOUGLAS
1995 Parallel Algorithms and Applications  
Time dependent partial di erential equations are often solved using algorithms which parallelize the solution process in the spatial domain. However, as the number of processors increases, the parallel e ciency is limited by the increasing communication computation ratio. Recently, several researchers have proposed algorithms incorporating time domain parallelism in order to increase e ciency. In this paper we discuss a class of such algorithms and analyze it rigrously. ; 2 ; : : : n ; 1 where
more » ... and B are m m matrices m is the number of grid points in the interior of the domain whose elements depend on L and B and u k is an m vector containing function values u at all the grid points at the k 0 th time step. Starting from u 0 , which is known from the initial condition, we can use an iterative algorithm such as Jacobi, Gauss-Seidel, or SOR to solve 1 sequentially for each time step: u k i = T u k i , 1 +c; i = 1 ; 2 ; : : : ; k= 1 ; 2 ; : : : n ; where T is the iteration matrix and c is a vector of known values. Usually, the entire process is spatially parallelized by splitting the domain into subdomains and distributing problems on the subdomains to multiple processors see 3 and 5 . At each iteration, the processors need to exchange boundary information with processors holding adjacent subdomains. As the number of processors increases, the communication computation ratio increases making the parallel e ciency decrease. In an e ort to forestall this and to allow increasing numbers of processors to
doi:10.1080/10637199508915498 fatcat:gfsbrey7ffhjrcqa6i5crfeuxu