Preconditioned conjugate gradient methods for boundary-domain integral method

Abstract

The paper deals with the iterative solution of systems of linear equations arising in viscous flow computation by the boundary-domain integral method (BDIM) with the subdomain technique. Three versions of conjugate gradient method — the bi-conjugate gradient method (bi-CG), conjugate gradients squared (CGS) and its variant bi-CGSTAB - are compared with the Gauss elimination direct method. Different types of preconditioning of matrices are tested including Jacobi and incomplete factorisation (ILU) preconditioners. A comparison of iterative and direct methods is done on a few test examples including Poiseuille's flow in a narrow channel and flow in a channel with circular obstacles. Special attention is given to behaviour of iterative methods in cases of fluid flows with higher Re numbers, where systems of linear equations become ill-conditioned. Whereas for low Re numbers all types of preconditioning used behave very well, this is not the case with higher Re numbers, where only ILU preconditioning preserves the stability and convergence of conjugate gradient methods. Among CG methods CGS and bi-CGSTAB are to be preferred since they reduce the error in the fastest and smoothest way. Computed test examples show that preconditioned CG methods can offer some major advantages over direct methods in the case of memory demands and computer time consumption.