Alternating Direction Implicit Method
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- Aubrey Weaver
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1 Alteratg Drecto Implct Method Whle dealg wth Ellptc Eqatos the Implct form the mber of eqatos to be solved are N M whch are qte large mber. Thogh the coeffcet matrx has may zeros bt t s ot a baded system. Of corse sometmes smplfcato ca be doe sg parttog of the matrx. Peacema ad Rachford 955 sggested a scheme kow as Alteratg Drecto Implct Method. I ths scheme the solto s obtaed two stages. Stage: The FD represetato of the Laplace Eqato (4. 4 s solved row-wse.e. we frst pt = ad =-----N gvg (4. 4 a system of N eqatos whch are solved by ay method. The same s coted wth = =3---- =M. Ths a system of N eqatos s solved M tmes. I geeral: ( ( Stage: I ths stage the solto proceeds vertcally.e. frst = =------M; = =------M ad so o. Here M eqatos are solved N tmes. Ths we have ( ( Ths stages ad are compted tll the accracy s satsfed. The above relatos are mplct bt wth proper orderg of the eqatos the coeffcet matrx becomes Trdagoal. The reslts obtaed eve mber of teratos are cosdered as the fal Sce the coeffcet matrces are always same so the comptatoal effort s less. Example: Solve the Heat Flow Problem a sqare of sze wth 4 / y x where x = ad x = s kept at C ad y= & y= are kept at C 5. Use ADI ad Lbema method to solve t ad compare the reslt.
2 Solto: By ADI Method: Gve Stage : The FD represetato of the Laplace Eqato 4 (4.3 Now fx = ; sbstttg = eqato (4.3 ad wth bodary codtos we have: O solvg these three eqatos we have: ; take take take ; assmed vale assmed vale assmed vale Now fx = ; sbstttg = eqato (4.3 ad wth bodary codtos we have: O solvg these three eqatos we have: ; 7.49 take take take ; assmed vale assmed vale assmed vale Now fx =3 ; sbstttg = eqato (4.3 ad wth bodary codtos we have: O solvg these three eqatos we have: Stage : 9 8 ; take take take ; assmed vale assmed vale assmed vale 3.486
3 The FD represetato of the Laplace eqato s : 4 (4.4 Now fx = ; sbstttg = eqato (4.4 ad wth bodary codtos we have: O solvg these three eqatos we ; 7.49 have: take take take ; Now fx = ; sbstttg = eqato (4.4 ad wth bodary codtos we have: O solvg these three eqatos ; 3 take we have: ; take take Now fx =3 ; sbstttg = eqato (4.4 ad wth bodary codtos we have: O solvg these three eqatos 3 ; we take take have: ; take As the vales at stage ad stage do ot match so we cote the process. Aga performg stage : Now fx = ; sbstttg = eqato (4.3 ad wth bodary codtos we have:
4 4 4 4 O solvg these three eqatos we ; 35. have: take take take ; Now fx = ; sbstttg = eqato (4.3 ad wth bodary codtos we have: O solvg these three eqatos we have: ; take take take ; Now fx =3 ; sbstttg = eqato (4.3 ad wth bodary codtos we have: O solvg these three eqatos we have: ; 35. take take 3 take ; Aga performg stage : Now fx = ; sbstttg = eqato (4.4 ad wth bodary codtos we have: O solvg these three eqatos 3 ; we have: ; take take take Now fx = ; sbstttg = eqato (4.4 ad wth bodary codtos we have:
5 4 4 4 O solvg these three eqatos we have: ; take take take ; ; ; ; Now fx =3 ; sbstttg = eqato (4.4 ad wth bodary codtos we have: take take take O solvg these three eqatos we have: 3 ; ; 3 Now reqred accracy s reached. Hece ths s the fal solto. 3 ; ; ; ; ; ; By Lbem Method: The FD represetato of the Laplace Eqato: 4 (4.5 Sbsttte = eqato (4.5 4 (4.6 Sbsttte = eqato (4.6 ad bodary codtos we have: Now sbstttg = ; = eqato (4.5 ad wth bodary codtos we have:
6 4 4 4 Now sbstttg =3 ; = eqato (4.5 ad wth bodary codtos we have: O solvg these e eqatoswe have: = ; =6.74 ; 3 = ; =3.5 ; 3 = =3.5 ; ; Comparg the two methods t ca be observed that the ADI method reqres less comptatoal efforts.
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