An efficient preconditioning method and its application to numerical simulation of steady and unsteady viscous flows

A time-accurate, fully implicit preconditioning method is developed and applied to solve a variety of steady and unsteady viscous flow problems. The space discretizaition is performed by employing a finite-volume cellcentred scheme and using a central difference with nearly 2nd-order accuracy. The time marching is based on a dual-time stepping method proposed by Jameson. LU-SGS (Lower- Upper Symmetric-Gauss-Seidel) implicit scheme within the framework of FAS (Full Approximation Scheme) multigrid method is applied to implement sub-iterations. A lowspeed matrix preconditioning is utilized to improve the efficiency and accuracy of simulating incompressible flows or low-speed flow regions in compressible flows. In order to obtain time-accurate solutions, only the pseudotime derivatives are preconditioned. Moreover, the original LU-SGS scheme and multigrid method are revised according to the preconditioned governing equations. The numerical examples show that the fully implicit preconditioning method is very efficient and robust for solving both 2-D and 3-D, steady and unsteady flow problems, especially for incompressible flow problems.