Efficient solution of fluid flow using the generalised conjugate grandient algorithm on a transputer-based machine

Abstract

The discretisation of the equations governing fluid flow gives rise to coupled, quasi-linear and non-symmetric systems. The solution is usually obtained by iteration using a guess-and-correct procedure where each iteration aims to improve the solution of the previous step. Each step or outer iteration of the process involves the solution of nominally linear algebraic systems. These systems are normally solved using methods based on the Gauss-Seidel iteration—such as the TDMA. However, these methods generally converge very slowly and can be very time consuming for realistic applications. In this paper, these equations are solved using the Generalised Conjugate Gradient (GCG) algorithm with a simple-to-implement Gauss-Seidel-based preconditioner on a distributed memory message-passing machine. We take advantage of the fact that only tentative improvements to the flow-field are sought during each iteration and study the convergence behaviour of the parallel implementation on a multi-processor environment.