5.2 Construction of Iteration series offorsolvingLinearAlgebraicEquations5.2.1Point(explicit)iteration5.2.2Block(implicit)iteration5.2.3Alternativedirection iteration-ADl11/56
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
5.2ConstructionofIterationMethodsofLinearAlgebraicEquations5.2.1 Point (explicit) iterationThe variable updating (更新) is conducted from node tonode; After every node has been visited a cycle (轮)ofiteration is finished: The updated value at each node isexplicitly related tothe others.1. Jakob iterationIn the updating of every node value the previous cyclevalues of neighboring nodes are always used;The convergencespeed is independent of iteration direction.2. Gauss -Seidel iteration12/56
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 always used; The convergence speed is independent of iteration direction. 1. Jakob iteration 2. Gauss－Seidel iteration
Always the present values (当前值) are used for updating3. SOR/SUR iteration (Successive over/under relaxation)α<1 Under-relaxation(k+1)d(k+1) = @(k) +α(Φ(K)(0≤α≤2)α > 1 Over-relaxationRemarks : 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 blockiteration is used, also calledimplicititerationImplicit means within each region all unknowns are solvedsimultaneously!13/56
13/56 3. SOR/SUR iteration (Successive over/under relaxation) ( 1) ( 1) ( ) ( ) ( ) k k k k + + = + − 1 Under-relaxation 1 Over-relaxation (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! Always the present values (当前值) are used for updating
2. Lineiteration(线迭代)-the most fundamental blockiterationThe smallest block is a line: At the same line TDMAis used for direct solution, from line to line iterative methodis used.Forexample:Solving in N-S direction and scanning (扫描) in E-W direction:Jakob:app(k) =avd(k+1) +asd(k+1) +[aΦ(k) +aw(k) +b](k+1)(k+1)asd(k+1) +[a(+awgW+ +b]G-S:apppaNOnANew b term, b扫描方向AScanning(扫描)inE-Wdirection14/56
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. For example: 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’
5.2.3Alternativedirectioniteration(交替方向迭代）-ADI1. 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 operationsof the entire domain consists of one iteration cycle :Alternative direction iteration (ADI)扫描方向-vs. alternative direction implicit (ADI)AIt canbe shownthat:one-time stepforward ofunsteady(transient)problem is equivalent to one cycle iterationforsteady problem15/56
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 operations 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