热流科学与工程西步文通大堂G教育部重点实验室0ay0f +ax2b2 +... + ak.k-Li9k-L1 + ak.kz+ibk-L1+1 +... +ak,k-1dk--+akd.+akk++..+ak+k +...+ak.LLixm =bkT1,1[bt,1T2.1b2,1主对角元系数值位置:TLi,1bul,1(L1 +1)个T1.2b1,2O元素...:9Ti.Mbi,M-atasauapdeT2,M1b2,M1::TLI-1,M1bL1-1,MTL1,M1LbL1.M1ΦCFD-NHT-EHT6/56CENTER
P aW a E a N a 1 1 1 , , ,2 2 , 1 1 , 1 1 1 , 1 , 1 1 , 1 1 , 1 1 , 1 1 1 . . . . k k k k L k L k k L k L k k k k k k k k k k k L k L k L M L M k a a a a a a a a a b S a 0 0 0 0 6/56
热流科学与工程西步文源大堂G教育部重点实验空Features of the ABEgs. of discretized multi-dimensionalflow and heat transfer problems:1) For conduction of constant properties in uniform grid-Thematrixis symmetricandpositivedefinite（对称、正定）；2) For other cases: matrix is neither symmetric nor positivedefinite.The ABEqs. of large scale sparse matrix（大型稀疏矩阵) are usually solved by iteration methods.5.1.2Directmethodand iterative methodforsolvingABEqs.1.Directmethod(直接法）Accurate solutioncanbeobtained viaafinitetimesofoperationsifthereisnoround-offerror（舍入误差),such.asTDMA,PDMAΦCFD-NHT-EI7/56CEN-F
Features of the ABEqs. of discretized multi-dimensional flow and heat transfer problems: 1) For conduction of constant properties in uniform grid— The matrix is symmetric and positive definite (对称、正定）; 2) For other cases: matrix is neither symmetric nor positive definite. The ABEqs. of large scale sparse matrix （大型稀疏矩 阵）are usually solved by iteration methods. 5.1.2 Direct method and iterative method for solving ABEqs. 1.Direct method(直接法) Accurate solution can be obtained via a finite times of operations if there is no round-off error (舍入误差), such as TDMA，PDMA. 7/56
热流科学与工程西步文源大堂G教育部重点实验空1（迭代法）2.Iterative methodFrom an initial field the solution is progressively(逐渐地improved via the ABEqs. and terminated (终止 when a pre-specified（预先设定）criterionissatisfied.The ABEqs. of fluid flow and heat transfer problemsusually are solved by the iteration methods :1) Due to the non-lineairity of the problems, the coefficients ofthe ABEgs need to be updated. There is no need to get the truesolution for the temporary（临时的）coefficients;2) The operation times of direct method is proportional to N2.5-3 ,where N is the number of unknown variables. When N is verylarge the operation times becomes very large, often unmanageable!CFD-NHT-EHT中8/56CENTER
From an initial field the solution is progressively(逐渐地) improved via the ABEqs. and terminated（终止） when a prespecified (预先设定) criterion is satisfied. 2. Iterative method（迭代法) The ABEqs. of fluid flow and heat transfer problems usually are solved by the iteration methods : 1) Due to the non-lineairity of the problems，the coefficients of the ABEqs need to be updated. There is no need to get the true solution for the temporary (临时的）coefficients; 2) The operation times of direct method is proportional to N2.5~3 ， where N is the number of unknown variables. When N is very large the operation times becomes very large, often unmanageable! 8/56
热流科学与工程西步文源大学教育部重点实验空5.1.3MaiorldeaandKeyIssuesofIterationMethods1.Major ideaIn the matrix form, the ABEqs. is :A@ = b . Its solution is@ =(A)-'b, (A)-'is the inverse matrix. Iteration method isto construct a series of gk in multi-dimensional space R (wherethedimensionofthespaceequalsthenumberofunknowns)such that-(k)when k→ -(A-(k(k-1)For the kth iteration Φ""= f(A,b, Φ"2.Keyissues of iteration methods1) How to construct the iteration series of &k ?2) Is the series converged?ΦFO-NHTEEH9/56CENTER
1. Major idea In the matrix form, the ABEqs. is ： A b . 1 ( ) , A b ( ) 1 k A b （ ） when k ( ) ( 1) ( , , ) k k f A b 2. Key issues of iteration methods 2) Is the series converged？ 5.1.3 Major Idea and Key Issues of Iteration Methods Its solution is Iteration method is For the kth iteration 9/56 1) How to construct the iteration series of k ？ 1 ( ) A is the inverse matrix. in multi-dimensional space R (where k to construct a series of the dimension of the space equals the number of unknowns) such that
热流科学与工程西步文源大堂E教育部重点实验室3)How to accelerate the convergence speed?5.1.4 Criteria for terminating (inner)iteration(1) Specifying iteration times ;代数方程选代求解（内选代）更新系数，推向下一层次(2) Specifying relative change非线性问题送代求解（外送代）ofvariablelessthanasmallvalue;(k)(k+)≤8:8 (k+1)b(k+l)+80maxmaxImax(3) Specifying the relative norm ofresidual（余量的范数）lessthanacertain small value. Detaileddiscussionwill be presented inChapter 6CFD-NHT-EHTG10/56CENTER
3) How to accelerate the convergence speed？ 5.1.4 Criteria for terminating (inner) iteration (1) Specifying iteration times； (3) Specifying the relative norm of residual (余量的范数）less than a certain small value. Detailed discussion will be presented in Chapter 6 (2) Specifying relative change of variable less than a small value； ( 1) ( 1) ( ) max max ; k k k ( 1) ( ) ( 1) 0 max k k k 10/56