Some Improvements On Krylov Subspace Methods And Their Applications For Soving N-S Equations

With the fast development technology,the solution of large sparse linear systems hasbecome the nucleus problem in sicentific and engineering computation,numerical simula-tion,finance optimization.Since its computation time takes large proportion of the wholeproblem,to solve large sparse linear systems efficiently can greatly improve the efficiencyof solving the whole problem.This thesis focuses on the Krylov subspace methods for solv-ing large linear systems,and proof on the convergence of TDMA algorithm in the iterativesolution of two-dimensional convection-diffusion problems.The main work is as follows:Firstly, the tri-diagonal matrix algorithm(TDMA) is usually adopted to solve the de-scretized equations of two or three dimensional convection-difusion equations by using finitevolume method, which satisfied boundedness and transitivity. The convergence of TDMA toiteratively solve the discretized equations of two or three dimensional convection-diffusionproblems is proved, and the sufficient condition of the convergence for the algorithm is ob-tained. In addition, numerical tests are presented to verify the effectivenes of the algorithm.The numerical results are compared with the Gauss-Seidel iterative algorithm.Secondly, an efficient polynomial preconditioner used for WGMRES is constructed,and a new algorithm is obtained. Numerical experiments exposes that the new algorithm canconsiderably reduce the iterative times and computation cost.Finally,the PWGMRES method is applied to solve the Navier-Stokes equation in fluiddynamics.The numerical tests for parallel sudden expansion tube is listed to verify the effi-ciency of the method.We also compared it with the WGMRES method.