2. ADI-line iteration is widely adopted in the numericalsolution of flow and heat transfer problemThe ABEqs. generated on structured grid system can besolved by ADI---each line has the same number of unknownsMM1i41LiML2i1Li16/56
16/56 The ABEqs. generated on structured grid system can be solved by ADI-each line has the same number of unknowns 2. ADI-line iteration is widely adopted in the numerical solution of flow and heat transfer problem
5.2.4ADl-iteration（交替方向迭代）isidenticaltoADl-implicit（交替方向隐式）ADI-iteration is identical to the ADI-Implicit of solvingmultidimensional unsteady problem for one time sten(1)ADI-Jakob iteration can be expressed扫描方向as:(2)b(k+1)b(h+1/2)k+1/2）(k+1/2) + α(k) +asd(k)+b]=0F(k+1/2)d(k+1)+asds(k+I)=anΦN(k+1)3(k+1/2) +b)0apPEThis expression is very similar toPeaceman-RachfordADImplicitmethodfor2Dtransientproblem:17/61
17/61 ADI-Jakob iteration can be expressed as：： ( 1/ 2) ( 1/ 2) ( 1/ 2) ( ) ( ) [ ] k k k k k P P E E W W N N S S a a a a a b + + + = + + + + ( 1/ 2) k b + ( 1) ( 1) ( 1) ( 1/ 2) ( 1/ 2) [ ] k k k k k P P N N S S E E W W a a a a a b + + + + + = + + + + This expression is very similar to Peaceman-Rachford ADImplicit method for 2 D transient problem： ( 1) k b + ADI-iteration is identical to the ADI-Implicit of solving multidimensional unsteady problem for one time step 5.2.4 ADI-iteration (交替方向迭代）is identical to ADI-implicit (交替方向隐式）
2-D Peaceman-Rachford methodtlDividing △t into two sub-periods.plicitAt/2n+1In the 1st sub-period △t /2X-direction is implicit, y-directionAt/2Atis explicit;-implicitIn the 2nd △t / 2 y-direction2-D AD Implicitis implicit, and x is explicit.(k+1/2)Letrepresent temporary values at middle time-24k,represent CD for2nd-order x-directionDderivative at time level k: then we have:18/61
18/61 2-D Peaceman-Rachford method Let represent temporary values at middle time ( 1/ 2) k + Dividing t into two sub-periods. In the 1st sub-period t / 2 x-direction is implicit, y-direction is explicit; In the 2nd t / 2 y-direction is implicit, and x is explicit. 2 , k x i j represent CD for 2nd-order x-direction derivative at time level k ; then we have: t 2 2-D AD Implicit
k+1/2Rd1st sub-+k+1/2+82)(1)ijperiod:△t / 2k+1/22nd sub-1(2)2 gk+1/2 + 820ktl)alo0period:△t / 2dt,j+-2dt, +dj-1k+1/2k+1/2.k+1/211Pi-1.ji+1.+k+1/2Eq.(1):Ar?△y2atadtaAtaA+1/2+1/2(1）+（o2AxAxAxAADImplicitapbae,anas,a(k+1/2)(k+1/2)(k+1/2)+[avd(k+asds+b]ADIteration+awwa.ΦpThe same can be shown for the 2nd sub-time period. Thus one-time stepforward of transient problem is equivalent to one cycle iteration for steadyproblem.19/61
19/61 1/ 2 , , 2 2 1 / , 2 , ( ) / 2 k i j k x y i j k i j k a i j t + + − = + 1st subperiod： (1) 2 nd subperiod： 1/ 2 , 2 1/ 2 2 , 1 , 1 , ( ) / 2 k i j k i j k i j k x i j y a t + + + + − = + (2) Eq.(1): 1/ 2 1/ 2 1/ 2 2 2 2 2 , 1, 1, , 1 , 1 , (1 ) ( )( ) ( )( ) (1 ) 2 2 k k k k k k i j i j i j i j i j i j a t a t a t a t x x x x + + + + − + − + = + + + + − P a , E W a a , S N a a b The same can be shown for the 2nd sub-time period. Thus one-time step forward of transient problem is equivalent to one cycle iteration for steady problem. 1/2 1/2 1/2 1/2 2 +1, , -1, , 2 -2 + = ; k k k k i j i j i j x i j x + + + + ( 1/ 2) ( 1/ 2) ( 1/ 2) ( ) ( ) [ ] k k k k k P P E E W W N N S S a a a a a b + + + = + + + + ADImplicit ADIteration 2 , 1 , , 1 , 2 -2 + = k k k k i j i j i j y i j y + −