FINITE DIFFERENCE SOLUTIONS TO THE TIME DEPENDENT HEAT CONDUCTION EQUATIONS. T T q 1 T + = (1)

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1 Recall he ime depede hea coducio equaio FINIE DIFFERENCE SOLUIONS O HE IME DEPENDEN HEA CONDUCION EQUAIONS + q () We have already ee how o dicreize he paial variable ad approimae paial derivaive. he ame echique may be applied o he ime variable. Le + + O ( ). () hi i a forward differece repreeaio of he ime derivaive. Coider a a eample, he oe-dimeioal coducio equaio i Careia geomery q + wih appropriae iiial ad boudary codiio. Wrie he coducio equaio i fiie differece form a ode "" ad ime level "" uig he forward differece repreeaio from Equaio o approimae he ime derivaive q + () (3) Equaio 3 may be olved direcly for he ew ime emperaure + o give F { } q + () + o α k (4) where he dimeiole group i called he Fourier Modulu. Fo α Noe: he oly ukow a ay paial ode i +. hi implie he oluio for he emperaure diribuio a each ime ep may be obaied by imply marchig hrough he paial grid evaluaig Equaio 4 a each poi. hi oluio give he emperaure profile a each ime ep eplicily i erm of kow iformaio from he previou ime ep. hee ype of oluio echique are herefore called Eplici Mehod. Coider ow he wo-dimeioal coducio equaio q + + y wih appropriae iiial ad boudary codiio. Agai wrie he coducio equaio i fiie differece form ad approimae he ime derivaive by a forward differece a ode "" ad ime "" o yield y y q () y (5) 4

2 Noe: Agai he oly ukow a each ime ep i he emperaure +. I hould be obviou, ha a imilar aalyi applied o a hree dimeioal problem would yield he ame reul. Eplici echique provide a relaively imple mehod for olvig mulidimeioal raie coducio problem. Numerical Sabiliy Ideally, he oluio o our fiie differece equaio hould coverge o he aalyic oluio of he correpodig parial differeial equaio a ad 0. Whe compuig a fiie differece oluio however, we geerae roud off error a each ime ep due o he fiie preciio ihere i all compuer. hee error ca do oe of hree hig: ) ed o cacel or damp each oher ou. ) Remai approimaely coa. 3) Carry over ad grow from oe ime ep o he e. hi la pheomea i kow a umerical iabiliy. I ca be how ha he abiliy of Eplici Mehod i a fucio of he Fourier Modulu. o preve umerical iabiliie i he Eplici Mehod dicued above, i ca be how ha he Fourier Modulu mu aify he followig abiliy crieria -D F o / -D F o /4 3-D F o /6 uder he aumpio of equal pacig i he, y ad z direcio. he Eplici Mehod are herefore aid o be codiioally able. I hould be obviou ha he abiliy crieria limi our elecio of ime ad paial ep ize. I addiio, a hee mehod are oly O( ) correc, mall ime ep are likely o be required o maiai reaoable accuracy, paricularly i rapid raie. While he ime ep ize dicaed by he abiliy crieria may o be overly rericive if oly fied ime ep are o be ued, i more ophiicaed ime advaceme cheme where he ime ep ize i allowed o icreae a he raie low or approache eady ae, hi may reul i ime ep which are far maller ha ha required by accuracy coideraio. Coider agai our oe-dimeioal coducio equaio, bu ow epre he ime derivaive hrough a backward differece repreeaio a ode "" ad ime "", i.e. + O ( ) (6) he fiie differece form of he coducio equaio i he + + q + () (7) or by hifig all ime level pecificaio by q + ( + ). (8) Noe: + ca o be olved for eplicily, bu i implicily a fucio of he emperaure a every ode o he compuaioal grid. A a reul, mehod of hi ype are uually referred o a Implici Mehod. Applicaio of 43

3 Equaio 8 reul i a ridiagoal yem of equaio which mu be olved a each ime ep i he imulaio. I ca be how ha hi mehod i ucodiioally able for all F o. I i however ill oly fir order correc ad a a reul accuracy will dicae he ulimae ime ep ize. If we ry o obai a O( ) correc mehod by uig a ceral differece repreeaio of he ime derivaive of he form + + O ( ) he he fiie differece form of he coducio equaio i + + q + () +. (9) Equaio 9 repree a ecod order correc, eplici, muli-ep mehod a he emperaure a he ew ime i give eplicily i erm of kow emperaure a wo previou ime ep. I ca be how however, ha hi mehod i ucodiioally uable for all F o. We would like o fid a mehod ha i O( ) ad ucodiioally able a he ame ime. CRANK-NICHOLSON ECHNIQUE Coider he ceral differece repreeaio of he ime derivaive a ode "" ad ime " + /" + / + + O( ). (0) For our oe-dimeioal eample, he fiie differece form of he coducio equaio i he + / q + ( + / ). () We eed o approimae / i erm of emperaure a ime ad + o be coie wih he ime derivaive ad avoid he iabiliy problem aociaed wih he muli-ep mehod preeed previouly. I addiio, hi approimaio hould be ecod order correc. If we approimae he paial erm a + / i.e., he average of he paial erm a ime ad +, i ca be how ha hi approimaio i ideed O( ) correc. he fiie differece form of he coducio equaio i he q + + ( + / ) + 44

4 he volumeric hea geeraio rae ca eiher be averaged over he ime ierval or evaluaed a ime + /. he mehod preeed above i kow a he Crak-Nicholo echique. he Crak-Nicholo echique i ecod order correc, implici, ad ca be how o be ucodiioally able for all F o. 45

5 ridiagoal mari equaio ake he form APPENDIX A. Soluio of ridiagoal Marice by he homa Algorihm a c b a c b a c b a c b a Applicaio of Gauia Elimiaio wih Back Subiuio o hi yem, akig io accou he zero, yield he followig recurive relaio which make up he homa Algorihm. ) α a For i,, αi ai bi( ci / αi ) ) g / α For i,, g ( b g )/ i i i i α i 3) g For i -,, i gi ( ci / i) i+ α Noe: he i ' are compued i revere order