Fourth Order Positively Smoothed Padé Schemes. for Parabolic Partial Differential Equations with. Nonlocal Boundary Conditions

Transcription

1 Applied Matheatical Scieces, Vol. 4, 200, o. 42, Fourth Order Positively Soothed Padé Schees for Parabolic Partial Differetial Equatios with Nolocal Boudary Coditios Mohaad Siddique Departet of Matheatics ad Coputer Sciece Fayetteville State Uiversity, Fayetteville, NC 2830, USA Abstract Parabolic partial differetial equatio with olocal boudary coditios arise i odelig of various physical pheoea i areas such as cheical diffusio, theroelasticity, heat coductio process, cotrol theory ad edicie sciece. This paper deals with the successful ipleetatio of the positively soothed Pade` schees (PSP) to two-diesioal parabolic partial differetial equatios with olocal boudary coditios. We cosidered both Hoogeeous ad Ihoogeeous cases. The uerical results show that these uerical schees are quite accurate. Keywords: Fourth order positively soothed Pade schees, parabolic partial differetial equatios, olocal boudary coditios. Itroductio Parabolic partial differetial equatios (PDEs) with olocal boudary coditios arise i the atheatical odelig of iportat applicatios i scieces [6, 7, 8, 9, 22]. I the past two decades, a uber of uerical ethods [, 7, 0, 7, 8,

2 2066 M. Siddique 9, 23] for the uerical solutio of parabolic PDEs with olocal boudary coditios have bee developed. Twizell et al. [] reported that soe of these ethods (explicit ethods) suffer stability restrictios. I this paper we cosider the ipleetatio of both hoogeeous positively soothed Padé (PSP()) ad ihoogeeous positively soothed Padé (IPSP()) schees for the uerical solutio of two-diesioal parabolic PDEs with olocal boudary coditios. The PSP() ad IPSP(), uerical schees of order 2 (where is a positive iteger), have recetly bee developed by Wade et al. [4, 5] ad applied to various exaples fro fiacial atheatics, especially pricig optios with osooth payoffs. This is the first applicatio of the PSP() ad IPSP() schees for the uerical solutio of parabolic PDEs with olocal boudary coditios ad are based o a cobiatio of positivity preservig Padé [4] ad diagoal Padé approxiats. We will give a brief descriptio of Padé approxiats i the ext sectio. 2. Padé Approxiats If P(x)ad Q(x)are polyoials of degree ad respectively, the P(x) is a Padé approxiatio to a fuctio f (x) eas that Q(x) P(x) + + f (x) = + O(x ) Q(x) (2.) z I [], the Padé approxiat R, ( z) to the expoetial fuctio f ( z) = e is defied as follows: Let P(z) R =,(z) Q(z) (2.2) where ( + j)!! P ( z) = ( z) j= 0 ( + )! j!( j)! (2.3) ad ( + j)!! Q ( z) = ( z) j= 0 ( + )! j!( j)! (2.4) z + + Satisfyig R, ( z) = e + O( z ) as z 0, (2.5)

3 Fourth order positively soothed Padé schees 2067 We will call R, ( z ) a (, ) Padé schee of order ( + ). Whe =, the (, ) Padé approxiats are kow as diagoal Padé approxiats ad are deoted by R, ( z ). The positivity preservig Padé schees are a relatively ew research area; they have captured the iterest of atheaticias ad scietists. I the past few years, uch attetio has bee devoted to the developet of positivity preservig schees ad the cocept of positivity has eerged proietly because it has bee foud to be a iportat factor i cotrollig spurious oscillatios. The cocept of usig positivity-preservig Padé schees has bee discussed i a uber of papers [3, 4, 5, 20, 2]. Defiitio: A uerical schee is called a positivity preservig schee if the graph of its stability fuctio stays above the x-axis ad approaches zero ootoically. The ( 02, ) Padé are positivity-preservig schees. For = 23,,, K, we have ( 0, ) Padé, ( 03, ) Padé, ( 05, ) Padé, K as positivity-preservig schees. Y (, ) - Pade (2, 2) - Pade (3, 3) - Pade 0.2 O 0 X Figure. Aplificatio sybols of the first three diagoal Padé approxiats of exp(-z).

4 2068 M. Siddique Y (0, ) - Pade (0, 3) - Pade (0, 5) - Pade O X Figure 2. Aplificatio sybols of three positivity-preservig Padé i. e. (0, ) Padé, (0, 3) Padé ad (0, 5) Padé. ka The (, ) Padé approxiatio of the atrix expoetial e is approxiated by ka e { Q( ka)} P( ka) R, ( ka) (2.6) where k is the tie step ad A is a tridiagoal atrix. ka The approxiatio of the atrix expoetial by the (2,2) Padé, deoted by R ( ) 2,2 ka yields the ethod ( ) ( ) v I ka k A I ka k A v = (2.7) The (0,3) Padé approxiatio to the atrix expoetial e R ( ) 0,3 ka yields ( ) e ka, deoted by v = I + ka + k A + + k A v (2.8) 2 6 The atrix A is a tridiagoal atrix. The uber of diagoals of A icreases with the powers of A. For exaple A is a five diagoal atrix, A is seve ad A is

5 Fourth order positively soothed Padé schees 2069 a ie diagoal atrix ad so ill-coditioig of the atrix A coes ito picture. The coditio uber of a atrix A deoted by cod( A) ad is defied by cod( A) = A A. (2.9) The coditio uber of a atrix easures the sesitivity of the solutio of a syste of liear equatios to errors i the data. It gives a idicatio of the accuracy of the results fro atrix iversio ad the liear equatios solutios. This ca also cause coputatioal difficulties ad ake the schees coputatioally less efficiet. Techiques that eploy partial fractio decopositio of ratioal fuctios hadle this difficulty very effectively. Gallopoulos ad Saad [] have used (, ) Padé (diagoal Padé) ad costructed parallel algoriths usig the factorizatios. Khaliq et al. [2] has used the diagoal ad subdiagoal Padé approxiatios i factored ad partial fractio fors. They have used partial fractio fors of diagoal ad subdiagoal Padé approxiatios to costruct the followig efficiet algorith. Algorith for hoogeeous case: Step. For i =, 2, K, q+ q 2, solve ( ka cii ) yi = v s. Step 2. Copute ( < ) q q+ q2 + = i i + i i i= i= q + v w y 2 Re( w y ) (2.0) Step 3. Copute ( = ) q q+ q2 + = s + i i + i i i= i= q + (2.) v ( ) v wy 2 Re( wy) Algorith for ihoogeeous case: Step. For i =, 2, K, q+ q2, solve ( ka cii ) yi = wivs + kwi j f ( ts + τ jk) for y i. Step 2. Copute ( < ) q q+ q2 + = i + i i= i= q + (2.2) v y 2 Re( y ) Step 2. Copute ( = ) q q+ q2 + = s + i + i i= i= q + (2.3) v ( ) v y 2 Re( y ) * j=

6 2070 M. Siddique Usig partial fractio decopositio techique, we ca write R ( ka) ad R ( ) 0,3 2,2 ka respectively as ( ) v w ( ka c I) + w ka c I ] v (2.4) + = [ 2Re ( ) v + = [ I 2Re w ka ci ] v (2.5) I the ext sectio, we will give a brief descriptio of PSP () ad IPSP () uerical schees. 3. Positively Soothed Pade Schees The PSP() ad IPSP() uerical schees are desiged by Wade et al. [4, 5] to take advatage of the positivity-preservig Padé schees. These uerical schees use two steps of positivity-preservig Padé followed by the diagoal Padé schees. For exaple for = 2, we have PSP(2) ad IPSP(2) schees of order 4 which use two steps of the (0, 3) Padé followed by the (2,2) Padé schees. We preset PSP() schee followed by the IPSP() schee. Hoogeeous Case: PSP() Wade et al. [4] itroduced Positively Soothed Padé (PSP()) schees for hoogeeous parabolic partial differetial equatios. The PSP() uerical schees are of order 2. For 0 < k k0 ad oegative iteger, let t = kad { v } be the uerical approxiatios for { } = 0 ut ( ) with v = 0 0 = v. Let be a positive iteger, ad p the uber of special startig steps. The faily of PSP() schees [4] is as follows: R0, 2 ( ka) v 0 < p; v+ = (3.) R, ( ka) v > p. For siplicity of otatio, r s is utilized for the startig schee R 0, 2 ( ka) ad r for the ai schee R,. The particular value of p which works best is ot kow. But i [4] uerical experiets as well as covergece results show that p is ever required to be larger tha 2 i the PSP faily. For = 2, we have PSP(2) uerical schee as follows: v + R0, 3( ka) v 0 2; = R2,2 ( ka) v > 2. (3.2)

7 Fourth order positively soothed Padé schees 207 v + w ( ka c I ) + w ( ka ci) v 0 2; 2Re 2 = I 2Re ( ka ci) + w v > 2. (3.3) where c = (Real Pole) c = i 2 w = w = i 2 PSP(2) schee uses 2 steps of (0,3) Padé schee followed by (2, 2) Padé schee. Usig this soothig criteria a secod order schee PSP(2) ca be writte as:. First two tie steps of (0,3) Padé schee ( ) v = ( ka c I) + w v w 2Re 2 ka ci 0 ( ) v2 = w( ka ci) + 2Rew2 ka ci v 2. Reaiig tie steps of (2,2) Padé schee ( ) v+ = I + 2Re w ka ci v, > 2. First two tie steps of (0,3) Padé schee are sufficiet to capture the spurious oscillatios of (2,2) Padé schee i PSP(2) schee. Ihoogeeous Case: IPSP() Wade et al. [5] also developed positively soothed Pade schees for ihoogeeous parabolic partial differetial equatios ad used the otatio (IPSP). The faily of IPSP () schees is as follows: R0, 2 ( ka) v + k Pi( ka) f ( t + τ ik) 0 < p; i= v+ = (3.4) R, ( ka) v + k Pi( ka) f ( t + τ ik) > p. i=

8 2072 M. Siddique The forula to obtai P i i [5] is s l j l l! ( z) τ ipi( z) = r( z), l+ i= ( z) j= 0 j! l = 0,,2, K, s. (3.5) where r = R0, 2 or R, respectively. For = 2, we have IPSP(2) uerical schee as follows: IPSP(2) schee uses 2 steps of (0,3) Padé schee followed by (2, 2) Padé schee. Usig this soothig criteria a secod order schee IPSP(2) ca be writte as:. First two tie steps of (0,3) Padé schee v = s y R( y2) where ( ) ka c I y = w v + kw f ( t +τ k ) + kw f ( t +τ k )ad ( ) s s 2 s 2 ka c I y = w v + kw f ( t +τ k ) + kw f ( t +τ k ) s 2 s 22 s 2 c = c = i 2 w = w = i 2 w = , w = i 2 w = , w = i τ = ad τ = Reaiig tie steps of (2,2) Padé schee v = s v + + s 2 R( y) where ka ci y = wv + kw f ( t +τ k ) + kw f ( t +τ k ) ( ) s s 2 s 2 c = i, w = i

9 Fourth order positively soothed Padé schees 2073 w = i, w = i 2 I the ext sectio, we will deostrate the ipleetatio of PSP(2) ad IPSP(2) (both schees are of order 4) o odel probles take fro the literature. 4. Nuerical Experiets We cosider two odel probles fro the literature [, 3, 6], for which exact solutios are kow. We apply PSP (2) ad IPSP (2) to these odel probles. The errors betwee the exact ad uerical solutios are show i the tables for each proble. The graphs of uerical ad soothed solutios are also show. Proble. (Ishak [3]) Cosider the two-diesioal diffusio proble 2 2 u u u = α + ; 2 2 t x y 0 < x, y<, t > 0 (4.) subject to the iitial coditio x uxy (,,0) = ( ye ), 0 x, 0 y (4.2) ad the boudary coditios t u(0, y, t) = ( y) e, 0 t, 0 y, + t u(, y, t) = ( y) e, 0 t, 0 y, x+ t ux (,0, t) = e, 0 t, 0 x, (4.3) ux (,, t) = 0, 0 t, 0 x, ad olocal boudary coditio x( x) t uxytdxdy (,, ) = 2( 4 ee ), x,0 y. (4.4) The exact solutio is give by uxy (,, t) = ( y) e x+ t (4.5)

10 2074 M. Siddique Table. Exact ad PSP(2) solutios of two-diesioal Diffusio Proble x y Exact Solutio PSP(2) Abs. Rel. Error e e e e e e e e e e e+000 Figure. Nuerical Solutio of (2, 2) Padé

11 Fourth order positively soothed Padé schees 2075 Figure 2. Soothig of (2, 2) Padé usig PSP(2) soothig techique Proble 2 Cosider the two-diesioal ihoogeeous diffusio proble 2 2 u u u t 2 2 = + e ( x + y + 4), t > 0, 0 < x, y < (4.6) 2 2 t x y The proble has osooth data with iitial coditio u( x, y, 0) = (4.7) ad the boudary coditios 2 t u(0, y, t) = + y e, 0 t, 0 y, u y t y e t y 2 t (,, ) = + ( + ), 0, 0, ux t xe t x 2 t (,0, ) = +, 0, 0, ux t x e t x ad olocal boudary coditio 2 t (,, ) = + ( + ), 0, 0, (4.8)

12 2076 M. Siddique 00 2 t uxytdxdy (,, ) = + e. (4.9) 3 t 2 2 The exact solutio isu( x, y,t ) = + e ( x + y ). (4.0) Table 2. Exact ad IPSP(2) solutios of the Ihoogeeous Diffusio Proble x y Exact Solutio IPSP(2) Abs. Rel. Error e e e e e e e e e e e+000 Figure 3. Nuerical Solutio of (2, 2) Padé

13 Fourth order positively soothed Padé schees 2077 Figure 4. Soothig of (2, 2) Padé by PSP(2) soothig techique I order to verify uerically whether the PSP (2) schee leads to higher accuracy, we ca evaluate the uerical solutio. Tables ad 2 show the exact solutio, the uerical results of PSP (2) schee, ad the error betwee the exact ad uerical solutios for various values of x ad y, whe t =. The graph of Padé (2, 2) shows spurious oscillatios, which are captured by PSP (2) schee. 5. Coclusios I this work, we have eployed PSP (2) ad IPSP (2) uerical schees of order 4 for the solutios of two diesioal diffusio equatios with olocal boudary coditios o four boudaries. The probles cosidered cosist of both hoogeous ad ihoogeeous cases. To verify the accuracy of these schees for parabolic probles with olocal boudary coditios, the errors betwee the exact ad uerical solutios are coputed. Nuerical results show that the PSP

14 2078 M. Siddique (2) ad IPSP (2) schees are efficiet ad provide very accurate results. We have deostrated with tie evolutio graphs coputatioal perforace for the two odel probles. These uerical schees have proise due to their efficiet ipleetatio i solvig higher degree of polyoial atrices that arise with Padé schees as well as the potetial to ipleet i parallel. Refereces [] A. B. Guel, W. T. Ag ad E. H. Twizell, Efficiet Parallel Algorith for the Two Diesioal Diffusio Equatio Subject to Specificatio of Mass, Iter. J. Coputer Math, Vol. 64, p (997). [2] A. Q. M. Khaliq, E. H. Twizell ad D. A. Voss, O Parallel Algoriths for Seidiscretized Parabolic Partial Differetial Equatios Based o Subdiagoal Padé Approxiatios, Nuer. Meth. PDE, 9, pp. 07 6, 993. [3] A. R. Gourlay ad J. Morris, The Extrapolatio of First Order Methods for Parabolic Partial Differetial Equatios, SIAM J. Nuer. Aal. 7 (980), [4] B. A. Wade, A.Q.M. Khaliq, M. Siddique ad M. Yousuf, "Soothig with Positivity-Preservig Padé Schees for Parabolic Probles with Nosoth Data, Nuerical Methods for Partial Differetial Equatios (NMPDE), Wiley Itersciece, V. 2, No. 3, 2005, pp [5] B. A. Wade, A.Q.M. Khaliq, M. Yousuf ad J. Vigo Aguiar High-Order Soothig Schees for Ihoogeeous Parabolic Probles with Applicatios to Nosooth Payoff i Optio Pricig" Nuerical Methods for Partial Differetial Equatios (NMPDE) V.23(5), 2007, [6] Cao, J. R. ad va der Hoek, J., Iplicit Fiite Differece Schee for the Diffusio of Mass i Porous Media, Nuerical Solutio of Partial Differetial Equatios (Edited by Noye J.) North Hollad, pp , 982. [7] Cao, J. R., Y. Li ad S. Wag, A Iplicit Fiite Differece Schee for the Diffusio Equatio Subject to Mass Specificatio, It. J. Eg. Sci. 28 (990),

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