Higher Order Difference Schemes for Heat Equation
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1 Available a p://pvau.edu/aa Appl. Appl. Ma. ISSN: Vol., Issue (Deceber 009), pp. 6 7 (Previously, Vol., No. ) Applicaions and Applied Maeaics: An Inernaional Journal (AAM) Higer Order Difference Scees for Hea Equaion Jianzong Wang Deparen of Maeaics and Saisics Sa Houson Sae Universiy Hunsville, TX 77 USA _jw@ssu.edu Received: July 0, 009; Acceped: Ocober 5, 009 Absrac In is paper, we consruc e eplici difference scees for e ea equaion wi arbirary ig orders. We also sow e validiy of e new scees by nuerical siulaions. Keywords: Hea equaion; eplici difference scees; nuerical soluions; ig-order scees MSC (000) No.: 65M06 65D5 5K05. Inroducion In is paper we sudy e consrucion of e eplici difference scees for e -D ea equaion DucD u () were u u( ) is eperaure, c 0 is e ea conduciviy, D and operaor for of () is D Te D cd () For a eperaure funcion u ( ) le E ( R) denoe e spaial ranslaion opera k Eu( ) u( ) We ave E E kz A Lauren polynoial of E k 6
2 6 Jianzong Wang A n n ae k k k is called a difference scee of order s N if n k A 0 k 0 s A difference scee of order approiaes D Since e differenial operaor self-conjugae, we are only ineresed in e syeric difference scee of order : D is n n n n k k k k 0 A a ( E E ) wi A () 0and A ( ) 0 () For a eperaure funcion u we define e eporal difference operaor ( 0) by u ( ) u ( ) u ( ) Tus, an eplici difference scee for e ea equaion () is c n u A u () c Le 0 be e consan uliple of e raio of e ie-sep o e square of e space-sep (TSR). Ten, c ( O( )) A difference scee () is said o ave order s N if c n s Ru ( ) u Au ( ) 0 O Wrie E E I (5) were I is e ideniy operaor. Te siples difference scee for () is c u u (6) wic as order and is sabiliy condiion is [Gerald Wealey (999), Ricyer and Moron (967)]. People are also ineresed in iger order difference scees. In Qian e al. (000), e auors proposed e following difference scee of order c 6 u I u (7) and sowed a wen e scee is sable. In is paper, we sall give a general forula for e consrucion of difference scees for () wi arbirary orders and sow e validiy of
3 AAM: Inern. J., Vol., Issue (Deceber 009) [Previously, Vol., No. ] 65 e forula by nuerical siulaions.. Consrucion of Difference Scees We sar our consrucion fro e eponenial epansion of n n n D n. c Applying e ea equaion () and recalling we ave n D n n n n To illusrae our eod, we firs consruc e difference scees for () wi order and, respecively. Te Taylor epansion of in (5) is n n D (8) n ( n) wic yields c n n n c D n n ( n) n c nd( n) n ( n) (n) i.e., c c D O (9) Te forula (9) sows a e siples scee (6) as order and i acieves order wen 6 To derive e difference scee of order we, replacing in (8) by, derive e ideniy n D ( n) n n n and se e scee o
4 66 Jianzong Wang c ( ) a b (0) were a and b are wo real nubers o be deerined. We ave k c k ( k ) k ( a b ) c ( a b ) D k k ( k) Le ab a b 6. () Ten, c a b c a b D k k ( k) k ( ) ( ) k k ( k) ( 6 ) 6 6 c a b D O 6 60 Te soluion of () is a b. Terefore, seing a and b we ave c 6 6 ( a b ) c D O 6 90 i.e., e difference scee c c () as order By, we ave c c c [ ] ( )
5 AAM: Inern. J., Vol., Issue (Deceber 009) [Previously, Vol., No. ] 67 c 6 I wic sows a e scee () is e sae as e scee (7) obained in Qian, e al. (000). To obain e sabiliy condiion for e scee (), we denoe E I and rewrie e scee () o e for of E I I () Le A( ) be represenaion of E in e Fourier doain. Ten, by (), A( ) cos cos sin 8 sin Te sabiliy condiion of e scee () is a A R ( ) wic leads e sabiliy condiion We now develop e difference scee for () wi an arbirary order. Assue a e difference scee of order as e for c a j j () j were e coefficien vecor a [ a a ] T is o be deerined. Recall a c k k k k ( k ) k a kk c j a j D k k ( ) j In order o obain a scee of order e real nubers a a ave o saisfy ( k) k k j aj k j k (5) Wrie V 9 9 9
6 68 Jianzong Wang and ( ) T b 6 Te ari for of Equaion (5) is V ab Since e Vanderonde ari V is inverible, Equaion (5) as e unique soluion b a V (6) wic yields a kk c j a j D O k ( ) j c ( ) i.e., e difference scee () wi a in (6) as order Te scee () can be rewrien o E I akk (7) k Le A( ) be represenaion of cos A a k k k E in e Fourier doain. Ten, and e sabiliy condiion of e scee () is a A ( ) Terefore, a sufficien condiion for e sabiliy can be obained by a 0k and a 0 As eaples, we use (6) o derive e difference scees of order 6 and 8, respecively. k R k k Eaple. Le a a a
7 AAM: Inern. J., Vol., Issue (Deceber 009) [Previously, Vol., No. ] 69 wic is e soluion of e linear syse a a 9a a a 9 a 6 a a 9 a 60 Ten, e difference scee c ( a ) a a as order 6 We selec in e following range so a a a a and [ ( aa a)] are nonnegaive, a ensures e sabiliy of e scee. Eaple. Le a a a a wic is e soluion of e linear syse a a 9a 6a a a a a a a a a a a a a Ten, e difference scee c ( a ) a a a
8 70 Jianzong Wang as order 8. To ensure e sabiliy of e scee, we selec in e range [090955]. Nuerical siulaions To validae our eoreical resuls, we sow a nuerical eaple in is secion. For coparison, we se e sae iniial condiion as in Qian, e al. (000) for e ea equaion (): u ( 0) sin( k ) and seek for e uni-periodic (wi respec o ) soluion of (). Te eac soluion is ck u ( ) e sin( k ) To apply our scees o e equaion, we le 0 be e space-sep and 0 be e ie-sep, were is cosen suc a N is an ineger. Te relaion of and is given by c were is cosen fro e range of e sabiliy. Le u ˆ be e nuerical soluion obained by e difference scee of order We easure e global error of e scee a n by N E ( ) ( i) u( i) N i 0 uˆ and sow e poinwise error by e discree funcion Er ( ) uˆ ( ) u( ) 0 N As poined ou in Qian, e al. (000), e aial global error E() is obained a 0 wic is independen of scees. Le n were 0 c wic yields 0 ck n round k 0 Ten, e aial error is obained afer n 0 ieraions of e scees. In our nuerical siulaions, we se k c Ten, e global aial error is obained afer 500 ieraions. Te following able presens e aial global errors for all scees of order 6 and 8 Order of scee 6 8 Maial global error.70e e e-0.56e-6
9 AAM: Inern. J., Vol., Issue (Deceber 009) [Previously, Vol., No. ] 7 Reark: Te aial global error of e scee of order 8 already coes up o e 6 acine epsilon 0e 6 Te able sows a e nuerical resuls ac e eoreical resuls very well. Te following figures sow e poinwise errors of difference scees of order and 6 respecively.
10 7 Jianzong Wang Acknowledgeen Te auor would like o ank e edior and, especially, e reviewers for eir valuable coens. Te researc is suppored by NSF Gran of e Unied Saes, DMS REFERENCES Gerald, C. F. and Wealey, P. Q. (999). Applied Nuerical Analysis, 6 ed., Addison Wesley. Qian, Y., Cen, H., Zang, R., and Cen, S. (000). A new four order finie difference scee for e ea equaion. Counicaions in Nonlinear Science and Nuerical Siulaion 5, Ricyer, R. D. and Moron, K. W. (967). Difference Meods for Iniial-Value Probles, nd ed., Wiley, New York.
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