The main purpose of this paper is to develop stable versions of some Krylov subspace methods for solving the linear systems of equations Ax = b which arise in the difference solution of 2-D nonstationary Navier-Stokes equations using implicit scheme and to determine a good value of the time step. Our algorithms are based on the conjugate-gradient method with a suitable preconditioner for solving the symmetric positive definite system and preconditioned GMRES, Orthomin(K), QMR methods for solving the nonsymmetric and (in)definite system. The performance of these methods is compared. In addition, we show that by using the condition number of the first nonsymmetric coefficient matrix, it is possible to determine a good value of the time step.

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