Rotated Krylov preconditioned iterative schemes in the solution of convection-diffusion equations

This paper presents the numerical solution of the two-dimensional convection-diffusion partial differential equation (pde) discretized by several group iterative schemes based on the centred and rotated (skewed) five-point finite difference discretizations, namely the explicit group (EG) and explicit decoupled group (EDG) methods, respectively [W.S. Yousif, D.J. Evans, Explicit group over-relaxation methods for solving elliptic partial differential equations, Mathematics and Computer in Simulation 28 (1986) 453-466; A.R. Abdullah, The four-point explicit decoupled group (EDG) method: a fast Poisson solver, International Journal of Computer Mathematics 38 (1991) 61-70]. The application of a modified 2 × 2 block factorization preconditioner applied to the linear systems that arise from these group iterative schemes is discussed. Several Krylov subspace methods, such as Bi-CGSTAB, GMRES and TFQMR will be used to solve the preconditioned systems. The experimental results show that the condition numbers of the coefficient matrices of the transformed preconditioned systems are substantially reduced and thus result in improved convergence rates. The comparative performance analysis between the group schemes also indicate that the preconditioned EDG with Bi-CGSTAB acceleration technique is able to gain the most efficiency amongst the schemes tested.