0O国a,+ax20+..+ak.k-0k-L1+ak,k+0k-L+1 +..+ak.k-1k-+akkdk+ak+lk++..+ akk+dk+ +.+ ak,LylLixm = bT1,1[bt,1T2.1b2,1主对角元系数值位置:Tui,1bui,1(L1+1)个T1,2b1,2O元素.:blatT1.M1b1,M1asfauapdeT2,M1b2,M1::TLI-1,M1bL1-1,MTL1,M1LbL1.M16/56
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
Features of the algebraic equations (ABEqs.) of discretized multi-dimensional flow and heat transfer problems:1) For conduction of constant properties without source in uniform grid:The matrixis symmetric and positive definite（对称、正定）；2) For other cases: matrix is neither symmetric nor positive definiteTheABEqs.oflargescalesparsematrix（大型稀疏矩阵）areusually solved by iteration methods5.1.2Direct methodand iterativemethod for solvingABEqs.1.Directmethod(直接法）Accurate solution can be obtained via a finite times of operationsif there is no round-off error(舍入误差),such as TDMA，PDMA.7/56
Features of the algebraic equations (ABEqs.) of discretized multidimensional flow and heat transfer problems: 1) For conduction of constant properties without source 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
2.Iterativemethod（迭代法）From 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）criterionis satisfied.The ABEqs. of fluid flow and heat transfer problemsusually are solved by the iteration methods :1) Due to the non-linearity of the problems, the coefficients of theABEqsneedtobeupdated（更新）.Thereisnoneedtogetthetrue solution for the temporary（临时的）coefficients;2) The operation times of direct method is proportional to N2.5-3where N is the number of unknown variables. When N is verylarge the operation times becomes very large, often unmanageable(难易处理）！（20241112)8/56
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-linearity of the problems，the coefficients of the ABEqs need to be updated（更新）. 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 （难易处理）! （20241112） 8/56
5.1.3MaiorldeaandKeyIssuesofIterationMethods1.MajorideaIn the matrix form, the ABEqs. is :A@ = b . Its solution isΦ=(A)-'b，(A)-is the inverse matrix(逆矩阵)The major idea of the iteration method is as followsConstructing a series of @k in multi-dimensional space R(where the dimension of the space equals the number of unknowns)such that when k →o, ()(A-lb-(k)(k-1For the kth iteration (*) = f(A,b,3(k-l)2.Keyissuesof iterationmethodsdk?1) How to construct the iteration series ofd2) Is the series converged?9/56
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 For the kth iteration 9/56 1) How to construct the iteration series of k ？ 1 ( ) A − is the inverse matrix（逆矩阵). in multi-dimensional space R k Constructing a series of (where the dimension of the space equals the number of unknowns) such that The major idea of the iteration method is as follows:
3) How to accelerate the convergence speed?5.1.4Criteria for terminating (inner)iteration(1) Specifying iteration times ;代数方程选代求解（内送代）(2)Specifyingrelativechange更新系数，推向下一层次of variable less than a small value;非线性问题选代求解（外选代）(k+1)<8≤ (k+1)dg(k+1) + 0PmaxImaxmax(3) Specifying the relative norm ofresidual（余量的范数）lessthanacertain small value. Detailed discussionwill be presented in Chapter 6.10/56
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