热流科学与工程西步文源大学G教育部重点实验室5.2ConstructionofIteration seriesofforsolvingLinearAlgebraicEquations5.2.1Point(explicit)iteration5.2.2Block(implicit)iteration5.2.3A/ternativedirectioniteration-ADCFD-NHT-EHTΦ11/56CENTER
11/56 5.2.1 Point (explicit) iteration 5.2.2 Block (implicit) iteration 5.2 Construction of Iteration series of for solving Linear Algebraic Equations 5.2.3 Alternative direction iteration－ADI k
热流科学与工程西步文源大学G教育部重点实验室5.2ConstructionofIterationMethodsofLinearAlgebraicEquations5.2.1 Point (explicit) iterationThe variable updating(更新)is conducted fromnode to node; After every node has been visited a cycle(轮)of iteration is finished; The updated value at eachnodeis explicitlyrelatedtotheothers.1. Jakob iterationIn the updating of every node value the previous cyclevalues of neighboring nodes are used; The convergencespeed is independent of iteration direction.2. Gauss - Seidel iteration中CFD-NHT-EHT12/56CENTER
12/56 5.2 Construction of Iteration Methods of Linear Algebraic Equations. 5.2.1 Point (explicit) iteration The variable updating (更新) is conducted from node to node; After every node has been visited a cycle (轮) of iteration is finished; The updated value at each node is explicitly related to the others. In the updating of every node value the previous cycle values of neighboring nodes are used; The convergence speed is independent of iteration direction. 1. Jakob iteration 2. Gauss－Seidel iteration
热流科学与工程西步文源大堂E教育部重点实验室Present values are used for updating3.SOR/SURiterationα <1 Under-(k+1)d(k+1) = (k) +α(d(0≤α≤2)α >10ver-Remarks: This relaxation is for solving the linear ABEqs.,Not for the non-linearity of the problem studied5.2.2Block(implicit)iteration（块隐式）1. Basic ideaDividing the solution domain into several regions, withineach region direct solution method is used, while from blockto block iteration is used,also called implicit iteration.Implicit means within each region all unknowns are solvedCER-NHT-EnSimultaneously!G13/56CENTER
13/56 3. SOR/SUR iteration ( 1) ( 1) ( ) ( ) ( ) k k k k 1 Under- 1 Over- (0 2) Remarks：This relaxation is for solving the linear ABEqs., Not for the non-linearity of the problem studied. 5.2.2 Block (implicit) iteration （块隐式） 1. Basic idea Dividing the solution domain into several regions, within each region direct solution method is used, while from block to block iteration is used，also called implicit iteration. Implicit means within each region all unknowns are solved simultaneously! Present values are used for updating
热流科学与工程西步文源大堂E教育部重点实验室2. Line iteration(线迭代)-the most fundamentalblock iterationThesmallestblockisaline:AtthesamelineTDMAis used for direct solution, from line to line iterative methodis used.Solving in N-S direction and scanning (扫描) in E-W direction :Jakob: app(k+1) =ax(k+) +asd(k+1) +[ae(k) +awk) +b]+ask+) +[as+awo+) +b]G-S: ap&+) =ano(*+1) +-ANew b term, b扫描方向SA(扫描)ScanninginE-WdirectionΦCFD-NHT-EHT14/56CENTER
14/56 2. Line iteration（线迭代)-the most fundamental block iteration The smallest block is a line: At the same line TDMA is used for direct solution, from line to line iterative method is used. Solving in N-S direction and scanning (扫描) in E-W direction： ( ) ( ) ( 1 1 1) ( ) ( ) [ ] k P P N N S S k k E k W W k E a a a a a b Jakob: ( ) ( ) ( ) ( ) 1 1 1 ( 1) [ ] k k k k P P N N S S E E W W k a a a a a b G-S: Scanning （扫描）in E-W direction New b term, b’
热流科学与工程西步文源大堂E教育部重点实验室5.2.3Alternativedirectioniteration(交替方向迭代)-ADl1. Basic ideaFirst conducting direct solution for each row(行) (orcolumn 列) ， then doing direct solution for each column(or row); The combination of the two updating of theentire domain consists of one iteration cycle :AAlternative direction iteration (ADI)扫描方向vs. alternative direction implicit (ADI)AIt canbe shown that:one-time stepforward of unsteady(transient)problem is equivalent to one cycle iterationforsteady problem.GCED-NHT-E15/56CENTER
15/56 5.2.3 Alternative direction iteration(交替方向迭代)-ADI 1. Basic idea First conducting direct solution for each row(行)（or column 列），then doing direct solution for each column （or row)；The combination of the two updating of the entire domain consists of one iteration cycle : Alternative direction iteration (ADI) vs. alternative direction implicit (ADI) It can be shown that: one-time step forward of unsteady (transient) problem is equivalent to one cycle iteration for steady problem