The Parallel Solution of ABD

Many commonly used numerical methods for the solution of Boundary Value Problems (BVPs) for Ordinary Di erential Equations (ODEs) contain signi cant components that can be parallelized easily, and recent investigations have shown that substantial speedups are attainable on machines with a modest number of processors. However, a stage in most of these methods is the numerical solution of an Almost Block Diagonal (ABD) system of linear algebraic equations. Several authors have proposed parallel algorithms for solving such systems; unfortunately most are potentially unstable or o er only limited parallelism. As a result, solving ABD systems has proven to be a bottleneck in the overall execution time of parallel BVP codes. In recent papers, Wright presents new stable parallel algorithms for solving ABD systems ([Wrig 90b] and [Wrig 91a]). Each algorithm attains the theoretically optimal speedup for the problem if enough processors are available, and each can be adapted for use on architectures with fewer than the optimal number of processors. The algorithms in x3 of this paper were discovered independently and are similar to those in [Wrig 90b] and [Wrig 91a]. Extensions to the basic algorithm are described which make better use of available processors during the decomposition stage in order to increase parallelism in the back-solve stage. A new algorithm based on eigenvalue rescaling is presented in x4 which can be up to twice as fast as the fastest algorithm in x3. Numerical experiments show that this new algorithm does not exhibit the instability inherent in some earlier schemes. In addition, some insight supporting the numerical evidence is given.