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