PRECONDITIONING TECHNIQUES FOR THE BICGSTAB ALGORITHM USED IN CONVECTION-DIFFUSION PROBLEMS

In this work, we investigate the techniques for computing the precondi tioning matrices used in the conjugate gradient method The physical pr oblem we consider is the convection-diffusion problem with high cell P eclet number, which is differenced by the third-order upwind-biased sc heme. The preconditioning matrix is computed by the traditional Incomp lete Lower-Upper (ILU) decomposition of the coefficient matrix or the coefficient matrix that is produced by the first-order upwind differen ce. Since the upwind schemes are one-sided-biased, the factorization e rror is greatly influenced by the arrangement of the variable in the l inear system. At high cell Peclet number, we can reduce the factorizat ion error significantly by ordering the variable according to the loca l 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.