Several examples of the Crank-Nicolson method for parabolic partial differential equations

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1 Academia Jornal of Scienific Researc (4: , May 03 DOI: p://dx.doi.org/0.543/asr ISSN: Academia Pblising Researc Paper Several examples of e Crank-Nicolson meod for parabolic parial differenial eqaions Acceped Marc, 03 ABSTRACT Hamza A. Isede Deparmen of Maemaical Sciences, Redeemer s Universiy, PMB 3005, Redempion ciy, Nigeria. E- afeisede@yaoo.com Te paper is concerned wi e Crank-Nicolson sceme being xaposed wi a classical meod sed in solving parabolic parial differenial eqaions. Te paper considers wo solion meods for parial differenial eqaions, one analyic and one nmerical (finie difference meod. Te finie difference approximaion (FTCS, e Crank-Nicolson sceme, was implemened on e diffsion eqaion in order o solve i nmerically. Te aim was o compare exac solions obained by a classical meod (specifically e separaion of variables meod, wi e approximae solions of e Crank-Nicolson meod. Te Crank-Nicolson solions were readily obained by e se of Malab programming codes. Te emperares a specific ime-seps were compared wi eir analyical resl conerpar. Te errors were fond o be 500 smaller for a redcion of a 5 in e ime sep of one of or model problems. Tese resls were ablaed and presened a e end of e paper. Key words: Crank-Nicolson, separaion of variables, finie difference, analyical solion, cenral difference. INTRODUCTION Te searc for e exac solion o many real-life problems as been an ancien qes since e ime of early scieniss. B nfornaely, ill e presen, no all problems can be solved exacly. Tis is becase of nonlineariy, complicaed geomery of domains, and oer forms of complicaions in some problems, wic makes exac solions impossible and naainable. In sc siaions e approximae solion sffices, wi e sal radeoff a is e error. Tese real-life problems are ofen modeled ino differenial eqaions. Te maemaical formlaion of mos problems in science involving raes of cange wi respec o wo or more independen variables, sally represening ime, ofen leads o a parial differenial eqaion. Problems involving ime as one independen variable someimes lead o parabolic parial differenial eqaions, e simples of wic is e diffsion eqaion, derived from e eory of ea condcion (Smi, 965. Te diffsion eqaion plays an imporan role in a broad range of pracical applicaions sc as flid mecanics. Only a limied nmber of special ypes of parabolic eqaion ave been solved analyically and e seflness of ese solions is frer resriced o problems involving sapes for wic e bondary condiions can be saisfied. In sc cases nmerical meods are some of e very few means of solion. According o Smi (965, of e nmerical meods available for solving differenial eqaions ose employing finie-differences are more freqenly sed and more niversally applicable an any oer. Tere are many ypes of finie difference meods, depending on e finie difference approximaion sed for e given differenial eqaion. Among ese meods are e explici and implici finie difference meods and eir modificaions. Many aors ave sed ese meods, finding em very sefl. (D For and Frankel, 953, modified e

2 Academia Jornal of Scienific Researc; Isede 064 simple explici sceme (FTCP, and proved a i is mc more sable an e simple explici case, enabling larger ime seps o be sed. Te Hopscoc sceme was firs inrodced by Gordon (965, and was generalized in a series of papers by Gorlay (977 and Gorlay and McGire (97. Te meod is flly explici b ncondiionally sable and s ime seps of any size may be sed. However, according o Briz (988, e accracy of e simple explici meod is barely improved pon. Feldberg (987 also poined o a e meod as disadvanages wen simlaions involve bondary singlariies. Gavagan and Rolle (990 sed e Alernaing Direcion Implici (ADI comparing i wi e Hopscoc meod for simlaing cronoamperomery a a micro disc elecrode and fond a ADI is more accrae for a given mes size. X and Zang (0 also sed e ADI o solve wo-dimensional cbic nonlinear Scrödinger eqaions and considered e meod mos effecive in accracy and compaion cos. Te Crank Nicolson sceme, wic is forward ime cenral space (FTCS, was de o Crank and Nicolson in 947. According o Kreyszig (993, e ime derivaive was replaced by forward difference in ime becase we ave no informaion for negaive a e sar. Te freedom o experimen wi any vale of r is one of e reasons e Crank Nicolson sceme was cosen for is sdy, even og small vales of r yield more accrae resls. Becase of is ncondiional sabiliy and ease of implemenaion in a comper no maer ow small r becomes, e Crank-Nicolson meod was e coice of nmerical meod wose solion will be compared wi a obained sing e analyical meod, e separaion of variables meod. Te comped resls (Isede, 004, obained sing Malab, were ablaed and compared wi eir analyical conerpar a e end of e sdy. Te errors were also ablaed and presened alongside. MATHEMATICAL PRELIMINARY Consider e following ea eqaion wi c-consan. c ( xx Tis eqaion is sally considered for x in some fixed inerval, say, 0 x l, and ime 0. One prescribes e iniial emperare (x,0 = f(x (f given and bondary condiions a x = 0 and x = l for all 0. We may assme c = and l =. Sc a ea eqaion and prescribed condiions are given as: 0 x, 0 ( xx (0, (, 0 (3 x, 0 x0.5 x (,0 ( x, 0.5 x (4 A finie approximaion of Eqaion ( wi in x-direcion and k in -direcion is: k i, i i, i (5 i, were x = i and = k (i = 0,,, ; = 0,,, Eqaion (5 can be simplified, wi r i, i i, i i, (6 sbsiing = 0 and k = 000 in becomes: i, i, 8i i, 0 (7 r k as: k r, Eqaion (6 Applying e finie difference approximaion of e given iniial and bondary condiions, we obain: 0 0 0, = 0,,,, 0 i 5, 5 i 0 i 0 i0 i 0 (8 i 0(0 (9 Crcial o e convergence of is meod is e condiion (Trner, 994 r k, implying k (0 THE CRANK NICOLSON SCHEME Condiion (0 is a andicap in pracice. Indeed, in order o aain sfficien accracy, we ave o coose small, wic makes k very small by Eqaion (0. Tis will make e

3 Academia Jornal of Scienific Researc; Isede 065 compaion excepionally lengy, as more ime levels will be reqired o cover e region. A meod a imposes no sc resricion as r = k/ was proposed by Crank and Nicolson in 947 (Smi, 965. Te idea of e meod is e replacemen of e differen qoien on e rig and side of Eqaion (5 by ½ imes e sm of wo sc difference qoiens a wo ime rows. Hence, insead of Eqaion (5 we ave:,,,,,, i i i i i i i i k ( Te resling eqaion wi r = k/ is: ( r r( ( r r( i, i, i, i i, i, ( Eqaion ( is e Crank-Nicolson sceme. If we divide e x-inerval 0 x l in Eqaion ( ino n eqal inerval, we ave n inernal mes poins per ime row. Ten for = and i =,,n, ( gives a sysem of n linear eqaions for e n nknown vales,,, n-, in e firs ime row in erms of e iniial vales 00, 0,, n0 and e bondary vales 0, n (= 0. Similarly for =, =, and so on; a is for eac ime row we ave o solve sc a sysem of n linear eqaions resling from Eqaion (. Wriing ( more generally as a marix eqaion: MU mb (3 Were e nknown U = +, e known concenraions b = and M, m are ri-diagonal marices of coefficiens defined as: ( r r 0 0, r ( r r 0, 0 ( 0 3, ; MU r r r r ( r n, APPLICATIONS AND RESULTS Example (Smi, 965 Consider e following linear eqaion. 0 x, 0 x (4 Sbec o e following iniial and bondary condiions: ( x,0 sin x; (0, (, 0 (5 Te analyic solion is obained sing e separaion of variables meod: Le U XT in Eqaion (4 erefore XT ' X '' T and T ' T X '' X now, eac side ms be a consan say, a is Yielding T' X '' and, T X T ' T 0 and X '' X 0 eir respecive solions are X acos x bsin x and T ce and e general solion is U( x, XT e Acosx Bsin x (6 Te bondary condiion U(0, 0 is applied o Eqaion (6 o obain: U( x, Bsin xe (7 and e iniial condiion U( x,0 sin x is applied o Eqaion (7 o obain e solion: U( x, e sinx (8 ( r r 0 0 r ( r r 0 0 ( 0 3 mb r r r r ( r n, Crank-Nicolson solion Te Crank-Nicolson approximaion of e problem and is associaed bondary and iniial condiions are given as: ( r i, r( i, i, ( r i r( i, i,

4 Academia Jornal of Scienific Researc; Isede 066 Table. Crank-Nicolson and analyical solion a k=0.00. x=0.5 Time C-N solion Analyical solion Error E sini i0 0 0 for i,,...,5; 0,,...,0 Taking 0. and k 0.00, we obain e resls o is problem wi Malab codes. A x = 0.5 and = 0.005, 0.006, and 0.007, we ave e emperare vales for e Crank-Nicolson solion, as well as eir corresponding Analyic solion vales, and eir corresponding errors (Table. U( x, e ( Acos x Bsin x ( Using e condiion U(0,=0 in Eqaion (, and U(0,=0 in e resling eqaion, we obain (wi 0 an inegral mliple of n x (, sin 0 n 50 U x Be ( Applicaion of e iniial condiion o Eqaion (, gives e final solion as; 9 3 x 8 3 U( x, 50e sin 0e sinx 0e sin4x (3 Example (Spiegel, 97 x 0 Saisfying e following condiions; (9 Crank-Nicolson solion Te Crank-Nicolson approximaion of e problem and is associaed bondary and iniial condiions are as given: ( r r( ( r r( i, i, i, i i, i, (0, 0, (0, 0, 3 x ( x,0 50sin 0sin x 0sin 4 (0 Te analyic solion is obained sing e separaion of variables meod: Leing U=XT in Eqaion (9, separaing e variables and eqaing i o a consan, we obain; T T X X ' 0; " 0 wose solions are T c e ; X c cos x c sin x 3 yielding (wi A = c c, B = c c 3;, 3 i 0 0; 0, 0; i0 50sin 0sin i 0sin 4i for i,,...,0; 0,,...,0 Taking 0. and k A x = 0. and = 0.008, 0.009, and 0.00, we ave e emperare vales for e Crank-Nicolson solion, as well as eir corresponding analyic solion vales, and eir corresponding errors (Table. Problem 3 (Trner, 994 (0 x x (4 Saisfying e iniial and bondary condiions,

5 Academia Jornal of Scienific Researc; Isede 067 Table. Crank-Nicolson and analyical solion a k= x=0. Time C-N solion Analyical solion Error Table 3. Crank-Nicolson and analyical solion a k=0.00 and k= x=pi/8 x=pi/8 Time C-N solion Analyical solion Error Time C-N solion Analyical solion Error ( x, 0 cos x ; (0, e ; (, e (5 Te analyic solion is obained sing e separaion of variables meod: Leing U=XT in Eqaion (4, separaing e variables and eqaing o a consan, we obain; T T X X ' 0; " 0 wose solions are respecively; T ce ; X acos x bsin x yielding U( x, e Acos x Bsin x (6 Using e bondary condiions, we obain; U( x, e cos x (7 Applicaion of e iniial condiion o Eqaion (7, gives e final solion as; U( x, e cos x (8 Crank-Nicolson solion Te Crank-Nicolson approximaion of e problem and is associaed iniial and bondary condiions are given as: ( r r( ( r r( i, i, i, i i, i, ( k ( k 0 e ; 8 e ; i0 cos( i for i 0,,...,8; 0,,...,0 Te resls obained are given in Table 3. A x /8 and / 3, resls for wo ime-sep sizes are presened Table 3. Firs wi k 0.00 and = 0.00, 0.004, 0.006; and e second wi k and = 0.0, 0.0, In bo insances, we ave e emperare vales for e Crank-Nicolson solion, as well as eir corresponding analyic solion vales, and e errors are also given in Table 3. CONCLUDING REMARKS Te effeciveness of e Crank-Nicolson sceme as a compaional meod is one of or conclsions in is paper. Specifically, e errors were fond o be abo 00 o 300 smaller for a redcion of a 5 in e ime sep. Or resls agree wi exising finding in lierares a smaller

6 Academia Jornal of Scienific Researc; Isede 068 ime-seps yield more accrae resls. Tis was observed wen e k-vale was redced and e errors compared, as sown in e error colmns of Table (3. Tis empaically sows a e vales of r (= c k/ ms be kep reasonably small for a close approximaion o e solion of e parial differenial eqaion. ACKNOWLEDGEMENT Te aor will like o acknowledge e aors wose works were cied in is paper. REFERENCES D For EC, Frankel SP (953. Condiions in e nmerical reamen of parabolic differenial eqaions. Ma. Comp. 7(43:35-5. Briz D (988. Digial simlaion in elecrocemisry. California: Springer- Verlag. pp Feldber SW (987. Propagaional inadeqacy of e opscoc finie difference algorim: e enancemen of performance wen sed wi an exponenially expanding grid for simlaion of elecrocemical diffsion problems. J. Elecroanal. Cem. (-:0-06. Gavagan DJ, Rolle DS (990. Correcion of bondary singlariies in nmerical simlaion of ime-dependen diffsion processes a nsielded disc elecrodes. J. Elecroanal. Cem. 95(:-4. Gordon P (965. Nonsymmeric difference sceme. SIAM J. Appl. Ma. 3: Gorlay AR (977. Spliing meods for ime-dependen parial differenial eqaions. In D.A.H. Jacobs, edior, Te Sae of e Ar in Nmerical Analysis, (pp UT, USA: Academic Press. Gorlay AR, McGire GR (97. General opscoc algorim for e nmerical solion of parial differenial eqaion. J. Ins. Mas. Appl. 7:6-7. Isede HA (004. A sdy of some common meods of solving parial differenial eqaions (Unpblised maser s esis. Ambrose Alli Universiy, Ekpoma, Nigeria. Kreyszig E (993. Advanced Engineering Maemaics. USA: Jon Wiley & Sons. pp Smi GD (965. Nmerical solion of parial differenial eqaions. London: Oxford Universiy Press. pp.9-8. Spiegel MR (97. Advanced Maemaics for Engineering and Science - Scam s Oline Series. New York: McGraw-Hill Book Co. pp Trner PR (994. Nmerical analysis. London: Te McMillan Press Ld. pp X Y, Zang L (0. Alernaing direcion implici meod for solving wo-dimensional cbic nonlinear Scrödinger eqaion. Comp. Pys. Commn. 83(5: Cie is aricle as: Isede HA (03. Several examples of e Crank-Nicolson meod for parabolic parial differenial eqaions. Acad. J. Sci. Res. (4: Sbmi yor manscrip a p://