Numerical methods for solving ABEgs. of 2-D problems(1)Penta-diagonalalgorithm(PDMA,五对角阵算法）(2)Alternative（交替的)-direction implicit(ADI,交替方向隐式方法）2.2-DPeaceman-RachfordADImethodDividing △t into two uniform parts;In the 1st t / 2 implicit in x direction,△t/2n+1and explicit in y direction;At/2AtIn the 2nd △t / 2 implicit in y direction,implicitandexplicit inxdirection11/58
11/58 (1) Penta-diagonal algorithm(PDMA,五对角阵算法) (2) Alternative (交替的)-direction implicit (ADI, 交替 方向隐式方法) 2. 2-D Peaceman-Rachford ADI method Numerical methods for solving ABEqs. of 2-D problems. Dividing t into two uniform parts； In the 1st t / 2 implicit in x direction, and explicit in y direction; In the 2nd t / 2 implicit in y direction, and explicit in x direction. t 2
Set u, the temporary(临时的) solutions at the firstsub-timelevelsS?Tn---CD scheme for 2nd derivative at n time level in xi.directionT", - 2T', +T"LJS?T"Axui.1st sub-2-D ADIa(S u.rnPtime level:△t / 2The solution of ui, can be obtained by TDMA△t/2n+1by taking 8,T", as b-term with known values at ntime levelAt/2Tn+12nd sub-Atu.4=a(Sui, + S,Tn+l)time level:implicit△t / 2Tn+In+ is solved by TDMA and is the solutionat time level of (n+1)12/58
12/58 Set ui,j the temporary(临时的)solutions at the first sub-time levels -CD scheme for 2 nd derivative at n time level in x direction 2 , n x i j T , , , 2 2 , ( / 2 n i j n x i i y i j j j u u T a T t 1 ） st subtime level: 2 nd subtime level: , , 1 , 2 , 1 2 ( ) / 2 n i j x y i j n i j n i j a t u u T T t 2 2-D ADI The solution of ui,j can be obtained by TDMA by taking as b-term with known values at n time level 2 , n y i j T is solved by TDMA and is the solution 1 , n Ti j at time level of (n+1). 1, , 1, 2 i j n n n T - 2T +T i j i j x 2 , n x i j T
3.3-DPeaceman-RachfordADlmethodDividing △t into three uniform parts; In the 1st △t / 3 implicit in x ,and explicit in y, z directions; In the 2nd and 3rd △t /3 implicit iny ,z direction, and explicit in x, z directions and x,y , respectively:Set uij.k, Vij.k the temporary(临时的) solutions at two sub-time levels1st sub-"A-T=a(8u a +8,Tu +8.Tux)time level:△t/3Vi.j.k - u'jk2nd sub-=a(8u,j.k +8,vi,j.k +8ui,j.k)△t / 3time level:Ta-Vl.j.k3rd sub-=a(8vi.j,k +8,vj,k + 8?Tutk)timelevel△t / 3The algebraic equations of 3D transient HC problem13/58
13/58 3. 3-D Peaceman-Rachford ADI method Dividing t into three uniform parts; In the 1st t / 3 implicit in x , and explicit in y, z directions; In the 2nd and 3rd t /3 implicit in y ,z direction, and explicit in x, z directions and x,y , respectively; Set ui,j,k , vi,j,k the temporary(临时的)solutions at two sub-time levels , , , , , , 2 2 2 , , , , ( ) / 3 i j n i j k n n x y i j k z i j k k i j k T a T T t u u 1 st subtime level: 2 nd subtime level: , , 2 2 , , , , , , , , 2 ( ) / 3 n i j k x i i j k j k z y i j k i j k u a t v u u v 3 rd subtime level: , , , 1 , , 222 , , , , , 1 ( ) / 3 n i j k n i j k i j k n i j k n x z y i j k T a v v T t v The algebraic equations of 3D transient HC problem
is updated for one time step by such ADI method:adopting TDMA three times in x,y,z direction respectivelyIt's obvious that this solution procedure is not fullyimplicit, and for 3D case the time step is limited byfollowing stability condition:11≤1.5a\t(NAxIf the time step is larger than the value specified by theabove eq, the resulted numerical solutions will be oscillating(振荡）. We call that the solution procedure is not stableMore discussion on the numerical stability will bepresented in Chapter 7.14/58
14/58 2 2 2 1 1 1 a t( ) 1.5 x y z If the time step is larger than the value specified by the above eq., the resulted numerical solutions will be oscillating (振荡）. We call that the solution procedure is not stable . It’s obvious that this solution procedure is not fully implicit, and for 3D case the time step is limited by following stability condition: More discussion on the numerical stability will be presented in Chapter 7. is updated for one time step by such ADI method: adopting TDMA three times in x,y,z direction respectively
Majornumericalmethods(concepts)introducedinthischapter1.Fullyimplicitscheme（全隐格式）oftransientproblem,whichcan（保证）stable and physically meaningful numerical solution;guarantee2.Harmonic mean（调和平均）for determination of interface conductivity;3.Unified coefficient expression by introducing a scaling factor (8x)。_ (Sx) (8x)Mep1and a nominal radius;4.Linearlization of sourceterm（源项线性化）by S=Sc+SpΦp，Sp≤O;5.Additional source term method (ASTM，附加源项法)for treating 2nd and3rd kinds of boundary conditions;6.TDMA（三对角矩阵算法）forsolvingalgebraicequation;7.General expression of discretized heat conduction eq.apT,=aeT+awTw+b=ZanTnb+bPhysical meanings ofae,awReciprocal ofthermalresistancebetweentwopoints,thermalconductance15/58
15/58 Major numerical methods (concepts) introduced in this chapter 1. Fully implicit scheme (全隐格式） of transient problem, which can guarantee （保证）stable and physically meaningful numerical solution; 2.Harmonic mean （调和平均）for determination of interface conductivity； 3.Unified coefficient expression by introducing a scaling factor and a nominal radius; 4.Linearlization of source term（源项线性化） by , 0 ; C P P P S S S S 5.Additional source term method (ASTM，附加源项法) for treating 2nd and 3 rd kinds of boundary conditions; 6.TDMA （三对角矩阵算法）for solving algebraic equation; 7.General expression of discretized heat conduction eq. P P E E W W nb nb a T a T a T b a T b Physical meanings of aE ,aW: Reciprocal of thermal resistance between two points, thermal conductance. ( ) ( ) ( ) e e e e E P x x x