Research On A Modified Preconditioned GMRES Algorithm And Its Applications

The solution of large sparse linear systems is the core of many numerical simulations in scientific problems and engineering calculations,and is usually the most time-consuming part of the calculation.Therefore,how to efficiently solve the large-scale coefficient matrix is the main problem.This article mainly discusses the solution and application of Krylov subspace method and TDMA(Tri-diagonal matrix algorithm)algorithm in the convection diffusion problem.The main work is as follows:1: The basic principle of GMRES(Generalized minimal residual algorithm)algorithm is introduced.A precondition factor is constructed by using GMRES itself,and then GMRES is normalized.Finally,the precondition factor is combined with the normalized GMRES to form a new improved algorithm.The improved algorithm reduces the number of iterations of the original orthogonalized GMRES algorithm,and meanwhile reduces the time and storage space needed for operation.By numerical example,the improved algorithm needs less computation than the original one.In addition,the improved algorithm is applied to the solution of fluid mechanics,and the improved algorithm is verified by taking the parallel protruding tube as an example,which proves the effectiveness and feasibility of the improved algorithm in practical application,and compares with the improved algorithm to show the superiority of the improved algorithm.2: In this article,with the GMRES algorithm after improving and TDMA algorithm in the application of convection diffusion problems.Firstly,the two-dimensional convectiondiffusion equation is discretized by finite volume method,in which different terms are discretized in different formats.The convection term 2 is in the upwind scheme,the diffusion term is in the second order central difference scheme,the time term is in the implicit scheme,and the source term is linearized.And it satisfies three important characteristics,namely conservation,boundedness and transportability.Taking two-dimensional heated plate as an example,the improved GMRES algorithm and TDMA algorithm were applied to solve the problem,and compared with Jacobi algorithm,the advantages of the improved GMRES algorithm and TDMA algorithm were illustrated.Finally,the precondition GMRES algorithm and TDMA algorithm are applied to the shizuishan section of the Yellow River in ningxia.