Liu Xiaoming, Lu Zhiming, Liu Yulu. Krylov Subspace Projection Method and ItsApplication on Oil Reservoir Simulation[J]. Applied Mathematics and Mechanics, 2000, 21(6): 551-560.
 Citation: Liu Xiaoming, Lu Zhiming, Liu Yulu. Krylov Subspace Projection Method and ItsApplication on Oil Reservoir Simulation[J]. Applied Mathematics and Mechanics, 2000, 21(6): 551-560.

# Krylov Subspace Projection Method and ItsApplication on Oil Reservoir Simulation

• Rev Recd Date: 1999-12-10
• Publish Date: 2000-06-15
• Krylov subspace projection methods are known to be highly efficient for solving large linear systems.Many different versions arise from different choices to the left and right subspaces.These methods were classified into two groups in terms of the different forms of matrix Hm,the main properties in applications and the new versions of these two types of methods were briefly reviewed,then one of the most efficient versions,GMRES method was applied to oil reservoir simulation.The block Pseudo-Elinimation method was used to generate the preconditioned matrix.Numerical results show much better performance of this preconditioned techniques and the GMRES method than that of preconditioned ORTHMIN method,which is now in use in oil reservoir simulation.Finally,some limitations of Krylov subspace methods and some potential improvements to this type of methods are furtherly presented.
•  [1] Saad Y.Krylov subspace methods for solving large unsymmetric linear systems[J].Mathematics of Computations,1981,37(155):105~126. [2] Saad Y,Schultz M A.Conjugate gradient-like algorithm for solving nonsymmetric linear systems[J].Mathematics of Computations,1985,44(170):417~424. [3] Saad Y,Schultz M A.GMRES:A generalized minimum residual algorithm for solving nonsymmetric linear systems[J].SIAM J Sci Comput,1986,7(3):859~869. [4] Brown P N.A theoretical comparision of the ARNOLDI and GMRES algorithms[J].SIAM J Sci Comput,1991,12(1):58~78. [5] Desa C,Irani K M,Ribbens C J,et al.Preconditioned iterative methods for homotopy curve trackling[J].SIAM J Sci Comput,1992,13(1):30~45. [6] Ern A,Giovangigli V,Keyes D E,et al.Towards polyalgorithm linear system solvers for nonlinear elliptic problems[J].SIAM J Sci Comput,1994,15(3):681~703. [7] 卢志明.油藏数值模拟中的一种新方法[D].硕士论文.上海:复旦大学,1993. [8] Tan L H,Bathe K J.Studies of finite element procedures-the conjugate gradient and GMRES methods in ADINA and ADINA-F[J].Computers &Structure,1991,40(2):441~449. [9] Frommer A,Glassner U.Restarted GMRES for the shifted linear systems[J].SIAM J Sci Comput,1998,19(1):15~26. [10] Paige C C,Saunders M A.Solution of sparse indefinite systems of linear equations[J].SIAM J Numer Anal,1975,12(4):617~629. [11] Sonneveld P.CGS,a fast Lanczos-type solver for nonsymmetric linear system[J].SIAM J Sci Statist Comput,1989,10(1):36~52. [12] Freund R W.A Transpoe-free quasi-minimal residual algorithm for non-Hermitian linear systems[J].Numer Maths,1992,14(2):470~482. [13] Parlett B N,Taylor D R,Liu Z A.A look-ahead Lanczos algorithm for unsymmetric matrices[J].Math Comp,1985,(44):105~124. [14] Freund R W,Nachtigal N M.QMR:a quasi-minimal residual methods for non-Hermitian linear systems[J].Numer Math,1991,(60):315~339. [15] Zhou L,Walker H F.Residual smoothing techniques for iterative methods[J].SIAM J Sci Comput,1994,15(2):297~312. [16] Van Der Worst H A.Bi-CGSTAB:a fast and smoothly convergence variant of Bi-CG for the solution of nonsymmetric linear systems[J].SIAM J Sci Comput,1992,13(2):631~644. [17] Chan T F,Gallopoulos E,Simoncini V,et al.A quasi-minimal residual variant of the Bi-SGSTAB algorithm for nonsymmetric systems[J].SIAM J Sci Comput,1994,15(2):338~347. [18] Ressel K J,Gutknecht M H.QMR smoothing for Lanczos-type product methods based on three-term recurrences[J].SIAM J Sci Comput,1998,19(1):55~73. [19] 陈月明.油藏数值模拟基础[M].山东:石油大学出版社,1989. [20] 胡家赣.线性代数方程组的迭代解法[M].北京:科学出版社,1991. [21] Saad Y.A flexible inner-outer preditioned GMRES algorithm[J].SIAM J Sci Comput,1993,14(2):461~469. [22] Kasenally E M.GMBACK:a generalised minimum backward error algorithm for nonsymmetric linear systems[J].SIAM J Sci Comput,1995,16(3):698~719. [23] Saunders M A,Simon H D,Yips E L.Two conjugate-gradient-type methods for unsymmetric linear equations[J].SIAM J Numer An al,1988,25(4):927~940.

### Catalog

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142