Fast Iterative Method-FIM: Application to the Convection- Diffusion Equations

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1 IN , Eglad, UK Joural of Iformato ad Computg cece Vol. 6, No.,, pp. - Fast Iteratve Method-FIM: Applcato to the Covecto- Dffuso Equatos H.aberNaaf,.A.Edalatpaah + Departemat of Mathematcs, Islamc Azad Uversty, P. Code 66, Laha, Ira Departmet of Mathematcs, Faculty of ceces, Gula Uversty, P.O. Box 5-9, Rasht, Ira (Receved July 9,, accepted eptember, Abstract. I ths paper we preset a ew algorthm for solvg lear systems by teratve method. I ths method the rate of covergece has bee mproved well. The results show that ths ew algorthm coverges faster tha AOR, OR, OR, etc.ad works wth hgh accuracy. The Ma dfferece betwee these methods wth the others s that uses affe combatos ad also less parameter. Besdes, we have show that a dverget process ca be coverges by usg FIM method. Fally the method s tested by some umercal expermets. Keywords: teratve method, spectral radus, AOR, OR, OR, Covecto-Dffuso equato. Itroducto The teratve methods for solvg a lear system are well kow ad some of them lke MAOR, AOR, OR ad OR, are very popular [-].I ths artcle a ew teratve method presets whch uses less parameter ad coverges faster tha the other methods. There are examples of dverget teratve methods, whch coverge whe usg ths ew algorthm. Let Axb (. Where A R & b, x R ad A s osgular. The basc teratve formula for (. s ( x M Nx + M b,,k (. Where x s a tal guess.f A s splt to AM-N, where M s osgular, the the basc teratve method for solvg (. s (..ths teratve process coverges to the uque soluto X A b for ay tal vector value x R f ad oly f ( M N < where BM - N s called terato B matrx.uppose dag(ai ad AI-L-U, Where L ad U are strctly lower ad strctly upper tragular part of A, respectvely. Now for Jacob teratve method MI, NL+U ad for Gauss-edel MI-L, NU. That s acob : x b + Lx ( + Ux ( gauss sedel x b + Lx + Ux I the ew method, whch we call FIM, ot oly the ew terato values are used, the prevous values also used for computg the ext values. I other words we have Ax b ( ( x b + L{( k x + kx } + Ux ; k R (. Dfto..[5] f s { x, x, K, x }, the the set of all affe combatos of s s defed as ( (. + Correspodg author. E-mal addresses: haaf@gula.ac.r H. aber Naaf, saedalatpaah@gmal.com (.A.Edalatpaah. Publshed by World Academc Press, World Academc Uo

2 H.aberNaaf, et al: Fast Iteratve Method-FIM : Applcato to the Covecto-Dffuso Equatos. Basc dea of FIM A( s { x x λ x ; λ ; λ R} (.5 ( ( I the Gauss edel process,.e. x b + Lx + Ux We would lke to mprove Ux ad substtute the better value stead. There are two suggestos for mplemetg the above goal:.statstcal method ( ( Usg varace ad stadard devatoux ca be chaged to U ( x + δ ( x where, δ (x s stadard devato. (Cofdece terval, statstcal tests.predcto method.. Two steps predctos (the predcto method wth two sub terato Ths process lke OR cotas two half teratos each step, but our method frst the better value ( of Ux provdes ad the mproved more,.e. x x b + Lx b + Lx + Ux + Ux.. Four steps predctos I ths process each teratos volves quarter teratos. e, ( x b + Lx + Ux x b + Lx + Ux (. x b + Lx + Ux x b + Lx + Ux Example. Let A The stadard teratve methods ad the FIM method are used for ths matrx. The spectral radus each case computed ad compared wth each other Table. As the results show the spectral radus of FIM s smaller tha the others whch meas faster covergece. Iteratve method ( Table shows the results of example. pectral radus Jacob.9859 Gauss-edel.976 ymmetrc Gauss- edel.96 Two steps FIM.956 Four steps FIM.97 To show a proof for the FIM method ad to descrbe why t works well, we cosder the followg: (. JIC emal for cotrbuto: edtor@c.org.uk

3 Joural of Iformato ad Computg cece, Vol. 6 ( No., pp - 5 ( It s easy to see that terato matrx Two steps predctos s B { } MN ( I L U ad for Four steps predctos { } B B MN ( I L U ;geerally we have M N {( I L U} Now we obta M,N follows: ( I L U( I L U M N (. ( I ( I L A( I ( I L A M N I M N ( I L A ( I L A( I L A M ( I L I A( I L ce Gauss-edel, MG.I-L { } { } M M I + N M (. G. G. G. { } & { } G. G. G. G. G. G. G. G. M I + N M M N I + N M N M N (.5 Also we ca to show that { ( ( } M M I + N M + L + N M (.6 G. G. G. G. G. ( Lemma.[6,Theorem.6].let AM-NM-N be two regular splttgs of A(.e. M, N,, where A-.If M M,the ( M N ( M N < Now by cosderg (,(ad referrg to[7, Theorem.]we have the followg Comparso theorem Theorem..let A - ad the Gauss-edel splt s regular, the ( B+ ( B L ( B < ( BG. < ( BJacob < Proof. By te-roseberg Theorem; see [6; Theorem.8; p: ] we have ( BG. < ( BJacob < To prove the other equaltes, we frst show that AM -N s a regular splttg.by(.we ca coclude that the splt s regular so wth respect to (.ad(.6we have M + M L Ad by (the theorem s true. Now we mprove the covergece wth affe combatos.. Two steps modfed FIM We mprove the teratos as follows. M ( + + (.7 x b Lx Ux.. Four steps modfed FIM I ths case we have ( x b+ Lx + U{( k x + kx }, k R JIC emal for subscrpto: publshg@wau.org.uk

4 6 H.aberNaaf, et al: Fast Iteratve Method-FIM : Applcato to the Covecto-Dffuso Equatos ( x b+ Lx + Ux x b+ Lx + Ux x b+ Lx + Ux x b+ Lx + U{( k x + kx }, k R (.8.5. Geeral FIM The teratve matrces two ad four steps method respectvely are: Where s ( I L U. Therefore B s{( k s+ ki} & B s k s + ks (.9 {( } {( B s k s ks },, k R + (. ( x b+ Lx + Ux M (. x b+ Lx + U{( k x + kx }, k R. ome umercal tests ad ts theoretcal aalyss Lemm. []. Let A be a cosstetly ordered matrx, ad assume that the egevalues μ of Jacob terato matrx (B L + U are real ad ( B <. The the optmal relaxato parameter w OR s gve by wopt + For ths optmal value we have ( B opt wopt. Lemma.. []. uppose the egevalues of teratve matrx of Jacob method s μ,,, K. Table. Optmum values for the AOR method, case detfcato Overreactos factor w Accelerato factor r pectral radus ( (a (b ( μ μ < μ < μ ( μ / < μ < μ / < ( μ > ( μ ~w ~w ~w ~w ~w μ ( μ ( μ ( + ( μ ( μ + ( μ ( μ + ( μ μ( μ μ ( μ ( + ( μ ~w ( < μ μ ~w They are all real ad μ <, the we defe: /( μ /( μ rw, ( [( ( ] / / L I wl r I + r w L+ ru (. JIC emal for cotrbuto: edtor@c.org.uk

5 Joural of Iformato ad Computg cece, Vol. 6 ( No., pp - 7 If λ, λ are the roots of (., [ μ, μ] λ [( r + rwμ ] λ+ ( r + ( w r rμ (. μ m μ m ax μ μ <. (. μ, (, +, r (, + ~ μ w, w /( + (, the the results of mmzg the spectral radus []are Table. I the ext dscusso ad examples, we use the parameters of ths Table for comparsos of the results. Note that the secod colum of ths Table s mportat ad descrbes the case of method. Example.. Let A The egevalues by Jacob method are,.89,-.699 ad -.6.so we have case ( from Table.there fore the optmum value of spectral radus OR ad AOR s.95.but FIM method we have the followg to results 5 Table. shows the result of example. K e- Example.. Let A The egevalues by Jacob method are.76,.,-.65,-.8. ce < ( μ.65 < μ.8& ( μ /.8767 < ( μ.9996.the we have the case (a Table.there for the optmum value for spectral radus of OR ad AOR s.657. But FIM method we have the followg results Table. hows the results of example. k e- Example.. Let A JIC emal for subscrpto: publshg@wau.org.uk

6 8 H.aberNaaf, et al: Fast Iteratve Method-FIM : Applcato to the Covecto-Dffuso Equatos The optmum value of spectral radus for OR s.6667 ad for AOR s.565 but FIM method we have the followg results Table5. hows the resuls of example. 5 K Example.. Let A Ths s the case ( Table ad the optmum value of spectral radus for OR s.6667 AOR s zero. But for FIM method for example wth (, k- spectral radus s.858e-7 ad for Example.5. Let A Note that ths s a arbtrary matrx ad the Jacob method for ths matrx has real ad also complex egevalues. The results are Ad for FIM we have Table6 shows the results of example.5 method w r Jacob.99 Gauss-edel.98 OR..9 AOR Table7. shows results for FIM example.5 k Dverget example JIC emal for cotrbuto: edtor@c.org.uk

7 Joural of Iformato ad Computg cece, Vol. 6 ( No., pp - 9 Applcato of statoary teratve methods such as Jacob, Gauss-edel, OR, OR, AOR, etc s easy but the ma problem these methods s that the process compare wth the ostatoary teratve methods such as GMRE,BICG,QMR,etc more beaks doe.we ca to see that FIM method work better tha the other teratve methods whe we have a dverget case statoary teratve methods Let A I ths example the methods lke Jacob, Gauss-edel, OR, OR, AOR ad etc; are all dverge. The spectral radus for all of them has the value grater tha, e.g, for Jacob ad Gauss-edel we respectvely have.89 ad.8. but FIM we have the followg results. Table8.results for dverget example 5 k Now we solve AXb,where 5.6 b ad A A6,(,k.8 By FIM Method.the exact soluto s x[,,,].ths example computed wth MATLAB7 Codes o a persoal computer Petum - 56 MHZ. Table9. umber of terato FIM method for dverget example Iteratve REMARK. I above examples, we obta optmum parameter k, ust by computer computatoal. The ma goal of ths work s that we uderstad the fudametal dfferet betwee our method ad other methods the optmal cases. but, the questo that could be cosdered as a ope problem s :how could we obta the optmal parameter k? Ths problem s mportat. but by choce of usual value for k,the results are also amazg (see secto5.we ca to say,whe Jacob terato matrx s coverge, the sg of k s egatve ad whe Jacob terato matrx s dverge, the k sg s postve(see A,A 6. I the followg secto we ca see that the applcato of our method the model of covectodffuso equato 5. Numercal expermets I the followg expermets,wthout cosderg optmal k ad usg a terval or usual value for JIC emal for subscrpto: publshg@wau.org.uk

8 H.aberNaaf, et al: Fast Iteratve Method-FIM : Applcato to the Covecto-Dffuso Equatos k we ca see that FIM works very well. Exprmet5..(Applcato to the two-dmesoal covecto-dffuso equato We cosder the two-dmesoal covecto-dffuso equato ( uxx + u yy + δ ux + τ u y f ( x, y Δu O the ut square doma Ω [,] [,], wth costat coeffcets, δ τ ad subect to Drchlet boudary codtos. dscretzato by a fve-pot fte dfferece operator leads to a lear system AXb where X ow deotes a vector a fte-dmesoal space ad A R. Wth dscretzato o a uform grd, usg stadard secod-order dffereces for the Laplaca, ad ether cetered or upwd dffereces for the frst dervatves., the coeffcet matrx has the form A trdagoal[ bi, trdagoal[ c, a, d], ei] b ( + ; c ( + γ ; a ; d ( γ ; e ( τ Where h ; γ δh are Reyolds umbers. Also, the equdstat step-sze h / s used the dscretzato ad the atural lexcographc orderg s employed to the ukows ad the rght-had sde satsfes b, h f, (x,y. For detals, we refer to [8,9]. Whe δ, τ we test ths equato for some teratve methods, we deote spectral radus of terato matrx, by method. for 6( [ A ] ad by computato of spectral radus of terato matrx Jacob,Gauss-edel, symmetrc Gauss-edel, OR,OR ad AOR,we have; Jacob.988 G..969 symmetrcg..97 w.9 OR.97 w.9 OR.96 w.9, r.8 AOR.979 I Fg. We ca see that the spectral radus of terato matrx FIM(fve- stepsfor dfferet values of k. The observato ca be further llustrated by the spectrum pctures plotted followg fgure. For ( [ A ] sem-logarthmc scale s gve the Fg., we ca see that spectrum of the terato matrces FIM (eght-stepad other methods. I ths expermet parameters are gve as the followg; W.9,r.8,k- JIC emal for cotrbuto: edtor@c.org.uk

9 Joural of Iformato ad Computg cece, Vol. 6 ( No., pp - Fg..spectral radus of FIM(fve-steps wth some values of K Fg..spectra of the terato matrx FIM ad dfferet teratve methods(for Exprmet. (Applcato to the three-dmesoal covecto-dffuso equato Cosder the three-dmesoal covecto-dffuso equato ( u xx + u yy + u zz + u x + u y + u z f ( x, y, z Δu O the ut cube doma Ω [,] [, ] [, ], wth Drchlet boudary codtos. Whe the seve-pot fte dfferece dscretzato, for example, the cetered dffereces to the dffusve terms, ad the cetered dffereces or the frst order upwd approxmatos to the covectve terms are appled to the above model covecto-dffuso equato, we get the system of lear equatos (. wth the coeffcet matrx A Tx I I + I Ty I + I I Tz Where A R ad the equdstat step-sze h / s used the dscretzato o all of the Three drectos ad the atural lexcographc orderg s employed to the ukows. JIC emal for subscrpto: publshg@wau.org.uk

10 H.aberNaaf, et al: Fast Iteratve Method-FIM : Applcato to the Covecto-Dffuso Equatos I addto, deotes the Kroecker product, ad T x, T y, ad T z are trdagoal Matrces gve by + h h Tx trdagoal[ (,, ( ] + h h Ty Tz trdagoal[ (,, ( ] For detals, we refer to [-]. For ( [ A ] ad by computato of spectral radus of terato matrx Jacob,Gauss-edel, symmetrc Gauss-edel, OR,OR ad AOR,we have; Jacob G symmetrcg. OR OR AOR.85 w.9.9 w w.9, r I Fg. We ca see that the spectral radus of terato matrx FIM(four-stepsfor dfferet values of k. Fg..spectral radus of FIM(four-steps wth some values of K Fg..spectra of the terato matrx FIM ad dfferet teratve methods(for 5 JIC emal for cotrbuto: edtor@c.org.uk

11 Joural of Iformato ad Computg cece, Vol. 6 ( No., pp - For 5( [ A ] s gve the Fg., we ca see that spectrum of the terato matrces FIM (fve-stepad other methods. I ths expermet parameters are gve as the followg; W.9,r.8,k- 6. Cocluso The teratve methods lke Jacob, Gauss-sedel, OR, etc are very popular for solvg the lear systems ad because the applcato of these methods s easy so they have bee used very ofte by researchs.the oly problem these methods s that the process compare wth the other methods more beaks doe. ths artcle we have cosdered ths pot ad developed the FIM method such that works fast wth hgh accuracy ad ot accrued by breakg steps. The mportat pots ths artcle are usg affe combatos ad less parameter. Fally we would lke to suggest usg FIM method stead of usg commo teratve methods also usg ths method as a base for precodtog processes. 7. Refereces [] D.M.YOUNG. Iteratve soluto of large lear systems. New York ad Lodo: Academc Press, 97. [] G.Avdelas ad A.Haddmos. optmum accelerated overrelaxato method a specal case. Math. Comp. 98, 6:8-87. [] Nkolas M.Mssrls, Davd J.Evas. O the covergece of some geeralzed precodtoed teratve methods. IAM J.NUMER.ANAL. 98, : [] [] H.aberNaaf,.A. Edalatpaah. ome Improvemets I PMAOR Method For olvg Lear ystems. J.Ifo. Comp.c., 6: 5-. [5] MURTY, K, G. Lear programmg. Joh Wley&os, 98. [6] R.. Varga. Matrx Iteratve Aalyss(secod ed. Berl: prger,. [7] M. Bez, D.B. zyld. Exstece ad uqueess of splttgs for statoary teratve methods wth applcatos to alteratg methods. Numer. Math.997, 76: 9. [8] H. Elma ad G. H. Golub. Iteratve methods for cyclcally reduced o-self-adot lear systems. Math. Comput. 99, 5: 67 7 [9] H. Elma ad G. H. Golub. Iteratve methods for cyclcally reduced o-self-adot lear systems II. Math. Comput. 99, 56: 5. [] C.Gref,J.Varah. Iteratve soluto of cyclcally reduced systems arsg from dscretzato of the threedmesoal covecto-dffuso equato. IAM J. c. Comput. 998, 9: [] C.Gref,J. Varah. Block statoary methods for osymmetrc cyclcally reduced systems arsg from threedmesoal ellptc equatos. IAM J. Matrx Aal. Appl. 999, : JIC emal for subscrpto: publshg@wau.org.uk