Preconditioning techniques for the BiCGSTAB algorithm used in convection-diffusion problems

In this work, we investigate the techniques for computing the preconditioning matrices used in the conjugate gradient method The physical problem we consider is the convection-diffusion problem with high cell Peclet number, which is differenced by the third-order upwind-biased scheme. The preconditioning matrix is computed by the traditional Incomplete Lower-Upper (ILU) decomposition of the coefficient matrix or the coefficient matrix that is produced by the first-order upwind difference. Since the upwind schemes are one-sided-biased, the factorization error is greatly influenced by the arrangement of the variable in the linear system. At high cell Peclet number, we can reduce the factorization error significantly by ordering the variable according to the local flow direction. This technique, when used with the ILU decomposition of the coefficient matrix produced by the first-order difference, is found to have steady convergence rate with no serious wiggles at high cell Peclet number.