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Abstract: When the implicit finite difference scheme and linearization technique in conjunction with the artificial compressibility method are employed to discretize the unsteady incompressible Navier–Stokes equations, in each pseudo-time step, a structured system of linear equations is obtained and needs to be solved numerically. In this paper, preconditioning techniques are applied to solve such linear systems. Numerical example is given to illustrate the effectiveness of the proposed method. We also find that a combination of the linearization technique with the preconditioned GMRES method is quite efficient for the implicit finite difference discretization of the incompressible Navier–Stokes equations. [Copyright &y& Elsevier]

Abstract: In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, we present block SSOR and modified block SSOR iteration methods based on the special structures of the coefficient matrices. In each step of the block SSOR iteration, we employ the block LU factorization to solve the sub-systems of linear equations. We show that the block LU factorization is existent and stable when the coefficient matrices are block diagonally dominant of type-II by columns. Under suitable conditions, we establish convergence theorems for both block SSOR and modified block SSOR iteration methods. In addition, the block SSOR iteration and AF-ADI method are considered as preconditioners for the nonsymmetric systems of linear equations. Numerical experiments show that both block SSOR and modified block SSOR iterations are feasible iterative solvers and they are also effective for preconditioning Krylov subspace methods such as GMRES and BiCGSTAB when used to solve this class of systems of linear equations. [Copyright &y& Elsevier]