EVALUATION OF PRECONDITIONED KRYLOV SUBSPACE METHODS FOR NAVIER-STOKES EQUATIONS

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1 The Seventh Asia-Pacific Confeence on Wind Engineeing, Novembe 8-12, 2009, Taipei, Taiwan EVALUATION OF PRECONDITIONED KRYLOV SUBSPACE METHODS FOR NAVIER-STOKES EQUATIONS Aishe Zhang 1,2, Ming Gu 3, Cuilan Gao 4, Weiui Liu 4, Deqian Zheng 5 1 Vice Peofesso, State Key Laboatoy fo Disaste Reduction in Civil Engineeing, Tongji Univesity, Shanghai , China, acclan@263.net 2 School of Civil Engineeing, Shandong Jianzhu Univesity, Ji nan , China 3 Pofesso, State Key Laboatoy fo Disaste Reduction in Civil Engineeing, Tongji Univesity, Shanghai , China, minggu@mail.tongji.edu.cn 4 Gaduate Student, School of Civil Engineeing, Shandong Jianzhu Univesity, Ji nan , China, clgao@sdjzu.edu.cn 5 Ph.D Candidate, State Key Laboatoy fo Disaste Reduction in Civil Engineeing, Tongji Univesity, Shanghai , China, zhengdeqian388@tom.com ABSTRACT The pupose of this wok is to compae the pefomance of some peconditioned iteative methods fo solving the linea systems of equations, fomed at each time-integation step of two-dimensional incompessible Navie- Stokes equations of fluid flow. The Navie-Stokes equations ae discetized in an implicit and upwind contol volume fomulation. The iteative methods used in this pape include BiCG, CGS and GMRES. These Kylov subspace methods ae equipped with incomplete LU factoization peconditioning. To incease the convegence ate, a scaling technique is used. Ou numeical esults show that the peconditioned GMRES method pefoms bette than othe peconditioned Kylov subspace methods. KEYWORDS: KRYLOV SUBSPACE METHODS PRECONDITIONING TECHNIQUE NAVIER-STOKES EQUATIONS SPARSE LINEAR SYSTEMS NON-SYMMETRIC MATRICES Intoduction Among the well-known iteative methods fo solving non-symmetic linea systems, Kylov subspace methods combined with suitable peconditiones ae the most pactical and popula and new techniques ae still emeging. Ove the past seveal decades, many effots ae focused on the development of efficient iteative methods deived fom Kylov subspace algoithms and some pecontiones [Nachtigal et al. (1992)]. When the non-symmetic poblems ae to be solved, some epesentative examples of Kylov subspace methods include the genealized minimal esidual method (GMRES) [Saad and Schultz (1986)] the conjugate gadient squaed method (CGS) [Sonneveld (1989)] the conjugate gadient method applied to nomal equations (CGN) [Hestenes and Stiefel (1952)] the bi-conjugate gadient stabilized method (Bi-CGSTAB) [Vost (1992)] and the othogonal minimization method (ORTHOMIN) [Vinsome and Othomin (1976)]. In addition, the ecent development of desktop computing towad paallel pocessing also motivated us to study Kylov subspace methods fo the simulation of fluid flow poblems. The pupose of this pape is the numeical study of the pefomance of some peconditioned iteative methods fo solving systems of linea equations with non-symmetic matices poduced by the contol volume discetizations of two-dimensional Navie-Stokes equations. Fo the model example of a lid-diven cavity flow, we compae the pefomance of the following iteative methods: bi-conjugate gadient method (BiCG)[Mikic and Mose

2 (1985)], CGS and GMRES. The peconditioning technique, incomplete LU factoization [Keshaw (1978), Koniges and Andeson (1987)], is also pesented. The above iteative methods ae all in conjunction with the peconditione. Thee ae othe types of iteative methods and peconditiones fo solving Navie-Stokes equations. Thei pefomance, howeve, will be a topic fo the futhe study. The pape is oganized as follows. The mathematical model and discetization method ae pesented in the following section. Next, the algoithms of iteative and peconditioning methods ae descibed. The numeical esults ae shown in the last pat. MATHEMATICAL MODEL AND COMPUTATIONAL PROCEDURE To demonstate the pefomance of diffeent peconditioned Kylov subspace methods, we conside a two-dimensional incompessible fluid flow in a lid-diven squaed cavity, which is the simplest simulation in hydodynamics. In this poblem, the lid is stated gadually fom est until it eaches the dimensionless velocity of 1, which is maintained aftewads. The govening equations ae discetized by the contol volume method. The discetization in time is implicit. The cental and upwind diffeences ae adopted to evaluate the diffusive and convective tems, espectively. The SIMPLE (Semi-Implicit Methods fo Pessue Linked Equations) algoithm [Patanke (1980)] based on a staggeed gid system is used fo the pessue-velocity coupling. The following lineaized system of algebaic equations in implicit fom is obtained: ap P = ae E + a W W + a N N + as S + S P (1) whee the subscipts, P, E, W, N, and S, epesent the cente gid point and its fou neighboing gid points, espectively. This type of notation is widely used in the numeical simulation of incompessible flows. a i ( i = P, E, W, N, and S ) is the coefficients, and S P epesents a souce tem. This equation can be witten in the matix fom: Ax = b (2) whee x is the unknown flow field vaiables, A is the coefficient matix which is lage and has a non-symmetic, spase stuctue. The set of Equ.(2) is to be solved by using the specified iteative methods descibed below. METHODS FOR SOLVING NON-SYMMETRIC LINEAR SYSTEMS Thee commonly used Kylov subspace methods ae pesented in this pape. Bi-conjugate gadient method (BiCG) The BiCG algoithm is poposed by Mikic and Mose (1985). The method attempts to ecusively constuct a suitable set of othogonal basis vectos. An implementation of BiCG can be ealized as follows. Algoithm 1 (Peconditioned BiCG) x 0 Choose an initial guess T Compute = b Ax 0 0, 0 = b A x0 Set p0 = 0, q0 = 0, s0=1020 Fo k=1, 2, until convegence 1 T 1 zk = M k 1, z k = ( M ) k 1 (M is the peconditioned matix). sk = zk, k 1 β k = s k s k 1 p k = z k + β kp k 1

3 q k = z k + β kq k 1 α k = s k q k, Ap k x k = x k 1 + α k p k k = k 1 α k Ap k If End fo k = k 1 k, k α k Aq k < ε, then stop iteation Conjugate gadient squaed method (CGS) The idea of the CGS method, poposed by Sonneveld (1989), is based on a polynomial T vaiant of the conjugate gadient algoithms. CGS can avoid the computation of A in biconjugate gadient algoithm (Bi-CG), which inceases the convegence speed consideably. Algoithm 2 (Peconditioned CGS method) Choose an initial guess x0 Compute 0 = b Ax Solve M = 0 0 Set p =, q 0 = 0, s 0 = 0 Fo k=0, 1, 2, until convegence 1 y k = M Ap k α k = 0, k 0, y k s k +1 = q k α k y k xk + 1 = xk + αk ( qk + sk +1 ) k + 1 = k α k A( q k + s k +1 ) Solve M j = j+ + 1, + 1 < ε If k k, then stop iteation β k = k, 0, +1 0 k Else: q k + 1 = k β ks k + 1 p k + 1 = q k β k ( β kp k + s k + 1 ) End fo It was poved in Ref.[Sonneveld (1989)] that each iteative step equies twice the mount of wok necessay fo oiginal CG but not moe than Bi-CG method[vost (1992)]. Genealized minimal esidual method (GMRES) Algoithm 3 (peconditioned GMRES(m)) Choose an initial guess x 0 Compute 0 = b Ax 0 β = Compute 0 2 v 1 = 0 β Fo k=0, 1, 2, until convegence Fo j=1, 2,, m 1 z = M v j w = A z

4 Fo i=1, 2,, j hi, j = w, v j w = w hi, jv j H m = { h i, j } End fo i h j+ 1, j = w 2 v j + 1 = w h j +1, j End fo j Fom the appoximate solution: 1 x k + 1 = x k + M Vm y m whee y m minimizes β e1 Hmy 2 Compute k + 1 = b Ax k + 1 If k+ 1, k+ 1 < ε, then stop iteation Else: β = k v1 = k +1 β End fo k Peconditioning and scaling It is known that peconditioning techniques play an impotant ole in the pefomance of the iteative methods. The ate of convegence of iteative pocess can usually be speed up by the peconditioning methods. The constuction of effective and efficient peconditiones is lagely poblem dependent. The speed of convegence stongly depends on the condition max min max min numbe k of the matix A, whee k = λ A λ A, λ A and λ A ae the maximum and minimal eigenvalues of matix A, espectively. The convegence speed is slowe when the condition numbe is lage. Cetainly, the best peconditione of the system Ax = b is the invese of A. Thus the solution is then obtained by only one iteation step, but this is not a pactical choice. Among all the peconditiones the incomplete LU factoization (ILU) is the most popula fo iteative methods both in symmetic positive definite matices and nonsymmetic cases. Gaussian elimination is equivalent to factoing the matix A as A=LU, whee L is a lowe tiangula matix and U is an uppe tiangula matix. The foundamental idea of ILU factoization is to peseve the spasity in the L and U factos by modifying Gaussian elimination to allow fill-ins at only a esticted set of positions in the LU factos. Incomplete LU factoization algoithm eads as follows. Algoithm 4 (Incomplete LU factoization) Fo i = 1, 2,, n Fo j = 1, 2,, n If 1 i, j n then ( i j) min, s = a l u ij ij ik kj k = 1 If i j l ij then = sij If i < j u ij then = s End if ij l ii

5 End fo j u ii = 1 Fo j = i+ 1, i+ 2,, n u ij = uij lii End fo j End fo i A scaling of the set of equations is used in the system. The appoach is to be defined as follows. s i = max a 1 j n i j, i =1,2,,n, i, j = 1,2,, n a ˆi j = ai j si b ˆ i = bi si, i =1,2,,n whee Aˆ = aˆi j and bˆ= b ˆi ae the esulting of scaling. NUMERICAL RESULTS We pefomed many computations on diffeent meshes in space: 16 16, 32 32, fo Reynolds numbe of 100 and All discetization schemes use unifom mesh size h in the x and y diections. The tests wee investigated to compae BiCG, CGS, and GMRES. All the iteative methods wee combined with ILU peconditioning and scaling techniques. Computing pogams wee coded in FORTRAN90 and un ( ) on a P5B266-C 7 ( 0) pesonal compute. The oute iteation loop has the stopping citeion of n The inne iteation loop fo the lineaized system etuns to the oute loop when the initial esidual in 2-3 nom was educed by a facto of 10. All the esults ae collected in Tables 1 and 2. Table 1. Computing efficiency of diffeent iteative methods (Re=100) Mesh size Method CPU time TOTLI NSTEP ANIT BiCG CGS GMRES(5) GMRES(10) BiCG CGS GMRES(5) GMRES(10) BiCG CGS GMRES(5) GMRES(10) Table 2. Computing efficiency of diffeent iteative methods (Re=1000) Mesh size Method CPU time TOTLI NSTEP ANIT BiCG CGS GMRES(5) GMRES(10) BiCG CGS GMRES(5) GMRES(10) In Table 1 the dependence of the CPU time (in second), the total numbe of iteations fo solving all the linea algebaic equations, TOTIT, the total time stepping, TOTST, and the

6 aveage numbe of iteations pe time step, ANIT=TOTIT/TOTST ae shown fo Re=100. In Table 2, the same values as in Table 1 ae shown fo Re=1000. Fom the two tables, it can be seen that the CPU time inceases quickly with the numbe of gid nodes and the Reynolds numbe. Between two GMRES(m) methods, GMRES(5) seems bette than GMRES(10) in tems of CPU time and iteation numbes fo two Reynolds numbe cases. The pefomance of GMRES(m) depends on the chosen value of m. It can also be found that CGS equie less computing wok than BiCG method, but GMRES(5) is the best of all. The elative aveage numbe of iteation RANIT=ANIT/ANIT(BiCG) fo Re=100 is compaed. By compaison, it is found that the iteative method GMRES, paticulaly GMRES(5), is bette than othe two methods. CGS equies less iteation numbes than BiCG to each the convegence.fom ou test poblem, we can outline that the peconditioned GMRES(5) is the most competitive method. Figue 1 shows the velocity pofiles fo u along vetical centeline and v along hoizontal centeline, obtained by using some of the above methods in this wok as well as the available esults of Ref.[Ghia (1982)]. It is demonstated by the figues that the esults ae in good ageement and ou methods ae coect a: Re=100 b: Re = a: Re=100 b: Re =1000 u b a v a b y x (a) u pofiles along vetical centeline (b) v pofiles along hoizontal centeline Fig.1 Compaison of velocity pofiles along the squae cavity centelines ( : pesent(cgs fo Re=100, GMRES(5) fo Re=1000), o: Ghia (1982)). The elative aveage numbe of iteation RANIT=ANIT/ANIT(BiCG) fo Re=100 is shown in Fig.2. By compaison, it is found that the iteative method GMRES, paticulaly GMRES(5), is bette than othe two methods. CGS equies less iteation numbes than BiCG to each the convegence. In Fig.3, the coesponding values of RANIT fo Re=1000 is also pesented. CGS pefoms well than BiCG but less well than GMRES. Figs.2 and 3 also indicate that when the Reynolds numbe inceases fom 100 to 1000, the RANIT diffeences between BiCG and CGS, and GMRES(m) methods decease. Conclusions In this aticle, thee iteative methods: BiCG, CGS and GMRES, ae used fo solving the linea system fo which the coefficient matix is spase and non-symmetic. All the algoithms ae peconditioned by incomplete LU factoization. Ou expeiments indicate that BiCG equied moe computing wok than the othes. CGS pefomed bette than CGS, but the numeical simulation showed that GMRES seemed to be the most competitive method.

7 RANIT Fig.2 Compaison of elative numbe of iteations fo diffeent methods 1: BiCG, 2: CGS, 3: GMRES(5), 4:GMRES(10) ( : 16 16, : 32 32, : 64 64, Re=100). RANIT Fig.3 Compaison of elative numbe of iteations fo diffeent methods 1: BiCG, 2: CGS, 3: GMRES(5), 4: GMRES(10) ( : 32 32, : 64 64, Re=1000). Acknowledgements Financial suppots fom National Natual Science Foundation of China( , , and ) and National Technology R&D Pogam of China (2006BAJ06B05) ae gatefully appeciated. Ou gatitude also is expessed to Shanghai Postdoctoal Foundation(06R214153), the National Postdoctoal Foundation of China ( ) and Shandong Jianzhu Univesity doctoal Foundation. Refeences Ghia,U., Ghia, K. N. and Shin C. T. (1982), High-esolutions fo Incompessible Flow Using the Navie-Stokes Equations and a Multigid Method. J. Comput. Phys., 48, Hestenes, M. R. and Stiefel, E. (1952), Methods of Conjugate Gadients fo Solving Linea Systems, J. Res. Nat. Bu. Standads, 49, Keshaw, D. S. (1978), The Incomplete Cholesky-Conjugate Gadient Method fo the Iteative Solution of Systems of Linea Equations, J. Comput. Phys., 26,

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