NumericalHeatTransfer（数值传热学）Chapter5SolutionMethodsforAlgebraicEquations(Chapter7inthetextbook)1/56
1/56 Numerical Heat Transfer (数值传热学) Chapter 5 Solution Methods for Algebraic Equations (Chapter 7 in the textbook)
5.1 IntroductiontoSolutionMethodsof ABEqs5.2Construction of IterationMethods of LinearAlgebraicEquations5.3ConvergenceConditionsandAccelerationMethodsforSolvingLinearABEqs.5.4BlockCorrectionMethod-PromotingConservationSatisfaction5.5MultigridTechniques-PromotingSimultaneousAttenuationofDifferentWave-lengthComponents2/56
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
5.1IntroductiontoSolutionMethodsofAlgebraicEqs5.1.1Matrixfeatureofmulti-dimensionaldiscretizedequation5.1.2Directmethod and iteration method forsolvingalgebraiceguations5.1.3Majorideaandkeyissuesofiterationmethods（终止）iteration5.1.4Criteriafor terminating3/56
5.1.1 Matrix feature of multi-dimensional discretized equation 5.1.2 Direct method and iteration method for solving algebraic equations 5.1 Introduction to Solution Methods of Algebraic Eqs. 5.1.3 Major idea and key issues of iteration methods 5.1.4 Criteria for terminating (终止）iteration 3/56
5.1Introductionto SolutionMethodsof ABEqs5.1.1Matrixfeatureof multi-dimensionaldiscretizedeguationofHTandFFproblemsFor 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=epe+awow+anp+ads+a,+apg+bFor a 2D case with L1 × Mlunknown variables, thegeneral algebraic equation of kth variable is:akp,+ak.2p2 +....+ak.k-L1pk-L1 +ak,k-L+1pk-L1+1+...+ak.k-1pk-+ak,kdk +ak,+1Dk+1 +...+ak,+LiDk+Li +...+ak,L1oMipL1oM1 =bk4/56
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:
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)ofthecoefficientsisconducted as shownNI(i,j+1)right, then the order()(At)of coefficientsin oneWIP-Eline are:S(rj-1)...O..aw,ap,a0..anL1a5/56
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