《数值传热学》研究生课程教学资源（课件讲稿）Chapter 5 Solution Methods for Algebraic Equations

热流科学与工程西步文源大学E教育部重点实验室NumericalHeatTransfer（数值传热学）Chapter5SolutionMethodsforAlgebraicEquations(Chapter7inthetextbook)oInstructorTao,Wen-QuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryofThermalScience&EngineeringXi'an Jiaotong UniversityInnovativeHarborofWestChina,Xian2022-0ct-18CFD-NHT-EHTΦ1/56CENTER

1/56 Instructor Tao, Wen-Quan Key Laboratory of Thermo-Fluid Science & Engineering Int. Joint Research Laboratory of Thermal Science & Engineering Xi’an Jiaotong University Innovative Harbor of West China, Xian 2022-Oct-18 Numerical Heat Transfer (数值传热学) Chapter 5 Solution Methods for Algebraic Equations (Chapter 7 in the textbook)

热流科学与工程西步文源大堂G教育部重点实验室5.1 IntroductiontoSolutionMethodsof ABEqs5.2Construction of IterationMethods of LinearAlgebraicEquations5.3ConvergenceConditionsandAccelerationMethodsforSolvingLinearABEqs.5.4BlockCorrectionMethod-PromotingConservationSatisfaction5.5MultigridTechniques-PromotingSimultaneousAttenuationofDifferentWave-lengthComponentsCFD-NHT-EHTΦ2/56CENTER

5.1 Introduction to Solution Methods of ABEqs 5.2 Construction of Iteration Methods of Linear Algebraic Equations 5.3 Convergence Conditions and Acceleration Methods for Solving Linear ABEqs. 5.4 Block Correction Method –Promoting Conservation Satisfaction 5.5 Multigrid Techniques –Promoting Simultaneous Attenuation of Different Wave-length Components 2/56

热流科学与工程西步文源大堂E教育部重点实验室5.1IntroductiontoSolutionMethods of ABEqs5.1.1Matrixfeatureofmulti-dimensionaldiscretizedequation5.1.2 Direct method and iteration methodforsolvingABEqs5.1.3Majorideaandkey issuesof iterationmethods5.1.4Criteria for terminating iterationΦCFD-NHT-EHT3/56CENTER

5.1.1 Matrix feature of multi-dimensional discretized equation 5.1.2 Direct method and iteration method for solving ABEqs. 5.1 Introduction to Solution Methods of ABEqs 5.1.3 Major idea and key issues of iteration methods 5.1.4 Criteria for terminating iteration 3/56

热流科学与工程西步文源大堂E教育部重点实验室5.1Introductionto SolutionMethodsof ABEqs5.1.1Matrixfeatureof multi-dimensionaldiscretized eguationofHTandFFproblemsFor 2-D, 3-D flow and heat transfer problems, thediscretized equations with 2nd order accuracy:2-D appp=aepe+awpw+anp+asds+b3-D app=aepe+awow+and+ads+a,+ag+bFor a 2D case with L1 × Mlunknown variables, thegeneral algebraic equation of kth variable is:ak.p,+ak.2p2 +....+ak.k-L1pk-L1+ak.k-L+1pk-L1+1+...+ak.k-1pk-+ak,kdk +ak,k+1Dk+ +..+ak,k+L1Pk+LI +...+ak,L1oMDL1oM1 = bkCFD-NHT-EHTΦ4/56CENTER

5.1 Introduction to Solution Methods of ABEqs 5.1.1 Matrix feature of multi-dimensional discretized equation of HT and FF problems For 2-D, 3-D flow and heat transfer problems, the discretized equations with 2nd order accuracy: 2-D P P E E W W N N S S a a a a a b           3-D P P E E W W N N S S F F B B a a a a a a a b               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                                     4/56 For a 2D case with L1 M1unknown variables, the general algebraic equation of kth variable is: 

热流科学与工程西步文源大堂E教育部重点实验室For 2-D problem with 2nd order accuracy there areonly five coefficients at the left hand side are not equalto zero, and the matrix is of quasi (准)five-diagonal, alargescalesparsematrix（大型稀疏矩阵）If the 1-D storageM1(L1,M1)ofthe coefficients isconductedasshown(i,j+1)Nright, then the orderG-(+,)(i+1D)ofcoefficientsinone-WIPEline are:S(r-1)-..O..aw,ap,a0..a,9L1x中CFD-NHT-EHT5/56CENTER

5/56 For 2-D problem with 2nd order accuracy there are only five coefficients at the left hand side are not equal to zero, and the matrix is of quasi (准)five-diagonal, a large scale sparse matrix (大型稀疏矩阵). If the 1-D storage of the coefficients is conducted as shown right，then the order of coefficients in one line are:      ,. . , , ,. . 0. 0 S W P E N a a a a a

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 pre￾specified (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