Design of Parallel Numerical Algorithms

Abstract

In this paper some techniques for exposing parallelism in a problem are surveyed and some new parallel numerical algorithms for the direct solution of linear systems presented and compared with the existing sequential methods.

Further, a new explicit iterative scheme is presented for the numerical solution of 2 point boundary value problems. From the usual central difference approximations to the differential operator, a tri-diagonal system of equations has to be solved. In this new approach, the matrix is split into component matrices G₁,G₂ and an iterative method formulated which is easily expressed in explicit form. By alternating this strategy on the grid points results in the Alternating Group Explicit (AGE) method which is analogous to the ADI method.

Finally, a new explicit method for the finite difference solution of parabolic partial differential equations is derived. The new method uses stable asymmetric approximations to the partial differential equation which when coupled in groups of 2 adjacent points (4 points for 2 dimensions) on the grid result in implicit equations which can be easily converted to explicit form and offer many advantages especially for use on parallel computers. A judicious use of alternating this strategy on the grid points of the domain results in new explicit parallel algorithms which possess unconditional stability.