热流科学与工程西青文交通大堂G教育部重点实验室Numerical methods for solvingABEgs.of2-D problems(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-DPeaceman-RachfordADImethodDividing At into two uniform parts;In the 1st △t / 2 implicit in x direction,At/2n+1oliciand explicit in y direction,△t/23AtIn the 2nd △t / 2 implicit in y direction,implicitand explicit in x direction.11/56
11/56 (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
热流科学与工程西幸交通大堂E教育部重点实验室Set ui, the temporary(临时的) solutions at the firstsub-timelevelsS?Tn---CD scheme for 2nd derivative at n time leveli.1in x directionuij -T"1 st sub-a(8ui, +8,Tn)2-D ADItime level:△t / 2tThe solution of u,, can be obtained by TDMAic△t/2n+1by taking S,T', as b-term with known values at ntime level△t/2Tn+12nd sub-2Atui.ia(8u. j.k + S,Tn+)time level:implicit△t / 2Tftl is solved by TDMA and is the solutionat time level of (n+1),12/56
12/56 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 2 , , , , 1 , ( ) / 2 n i j n i n x y k i j i j j u u T a 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)
热流科学与工程西青文交通大堂E教育部重点实验室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.jik2nd 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 problem13/56
13/56 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
热流科学与工程西幸交通大堂E教育部重点实验室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(NA1AxIf the time step is larger than the value specified by theabove eq:, the resulted numerical solutions will be oscillatingWe call that the solution procedure is not stableMore discussion on the numerical stability will bepresented in Chapter7.14/56
14/56 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 . 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
热流科学与工程西幸交通大堂C教育部重点实验室Majornumericalmethods(concepts)introducedinthischapter1. Fully implicit scheme of transient problem, which can guarantee stable andphysically meaningful numerical solution;(8x)(8x), (8x)2.Harmonic mean for determination of interface conductivityAPTE3.Unified coefficient expression by introducing a scaling factor and a nominalradius,4.Linearlization of source term by S = Sc + Spdp, Sp ≤0;5.Additional source term method (ASTM) for treating 2nd and 3rd kinds ofboundary conditions,6.TDMA for solving algebraic equation,7.General expression of discretized heat conduction eqapT,=aT+awTw+b=ZanTn+bPhysical meanings of ae,aw.Reciprocalofthermalresistancebetweentwopoints,thermal conductance15/56
15/56 Major numerical methods (concepts) introduced in this chapter 1. Fully implicit scheme of transient problem, which can guarantee stable and physically meaningful numerical solution; 2.Harmonic mean for determination of interface conductivity 3.Unified coefficient expression by introducing a scaling factor and a nominal radius; 4.Linearlization of source term by , 0 ; C P P P S S S S 5.Additional source term method (ASTM) for treating 2nd and 3rd kinds of boundary conditions; 6.TDMA 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