Chebyshev Gauss Lobatto Pseudo spectral Method for One Dimensional Advection Diffusion Equation with Variable coefficients

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1 Sohag J Math 3, o, Sohag Journal of Mathematcs An Internatonal Journal Chebyshev Gauss Lobatto Pseudo spectral Method for One Dmensonal Advecton Dffuson Equaton wth Varable coeffcents Galal I El Baghdady and M S El Azab Department of Engneerng Physcs and Mathematcs, Faculty of Engneerng, Mansoura Unversty, El Gomhera St, Mansoura, Daahla, 3556, Egypt Receved: Aug 04, Revsed: Sep 05, Accepted: 8 Sep 05 Publshed onlne: Jan 06 Abstract: In ths paper, we present a Chebyshev pseudo spectral method based on a Chebyshev Gauss Lobatto zeros wth the ad of the Kronecer product formulaton for solvng one dmensonal parabolc advecton dffuson equaton wth varable coeffcents subect to a gven ntal condton and boundary condtons Frst, we ntroduce an approxmaton to the unnown functon by usng Chebyshev dfferentaton matrces and ts dervatves wth respect to space x and tme t Secondly, we convert our problem to a lnear system of equatons to unnowns at the collocaton ponts, then solve t Fnally, two examples are gven to llustrate the valdty and applcablty of the proposed technque wth the ad of L -norm error and L -norm error to the exact soluton A comparson between the presented method has been done wth cubc B-Splne fnte dfference method Keywords: One-dmensonal parabolc partal dfferental equaton, Chebyshev Pseudo spectral method, Chebyshev Dfferentaton matrces, Kronecer product Introducton The combnaton of advecton and dffuson s mportant for mass transport n fluds It s well nown that the volumetrc concentraton of a pollutant, ux, t, at a pont xa x b n a one-dmensonal movng flud wth a speed qx and dffuson coeffcent px n x drecton at tme t t 0 s gven by the one dmensonal tme dependent advecton dffuson equaton of the form u t + qx u x u px = fx,t, x wthx, t [a, b] [0, T], subect to the ntal condton and the boundary condtons ux,0=u 0 x, x [a,b], ua,t = g t, ub,t = g t, t [0,T], 3 where fx,t, u 0 x, g t and g t are nown functons and assumed to be smooth functons Whereas u s the unnown functon ote that px and qx are consdered to be postve and smooth functons quantfyng the dffuson and advecton processes, respectvely One dmensonal verson of the partal dfferental equatons whch descrbe advecton dffuson of quanttes such as mass, heat, energy, vortcst, etc [, ] Equaton has been used to descrbe heat transfer n a dranng flm [3], water transfer n sols [4], dsperson of tracers n porous meda [5], the ntruson of salt water nto fresh water aqufers, the spread of pollutants n rvers and streams [6], the dsperson of dssolved materal n estuares and coastal seas [7], contamnant dsperson n shallow laes [8], the absorpton of chemcals nto beds [9], the spread of solute n a lqud flowng through a tube, long range transport of pollutants n the atmosphere [0], forced coolng by fluds of sold materal such as wndngs n turbo generators [], thermal polluton n rver systems [], flow n porous meda [3] and dsperson of dssolved salts n groundwater [4] Many authors deal wth equaton numercally, but wth constant coeffcents, n whch qx = β and px = α For example, n [5] the authors used cubc Correspondng author e-mal: amoun973@yahoocom c 06 SP atural Scences Publshng Cor

2 8 G I El Baghdady, M S El Azab: Chebyshev Gauss Lobatto Pseudo Spectral B-splne collocaton method to fnd numercal soluton to problem wth constant coeffcents β, α The method of the fourth-order compact fnte dfference scheme was presented n [6] For the nonlnear case, the newell wghtehead segel type equatons [7], they also use cubc B-splne collocaton method In recent years there has been a hgh level of nterest of employng spectral methods for numercally solvng many types of ntegral and dfferental equatons, due to ther ease of applyng them for fnte and nfnte domans [8,9,0,,] The speed of convergence s one of the great advantages of spectral method Besdes, spectral methods have exponental rates of convergence; they also have hgh level of accuracy From the overvew of spectral approxmaton to dfferental equatons, the spectral methods have been dvded to four types, namely, collocaton [3, 4], tau [5, 6], Galern [7, 8], and Petrov Galern [9, 30] methods In the present contrbuton, whch s an extenson to the wor uses Legendre bass n [3], we construct the soluton usng the pseudo spectral technques [3, 33] wth Chebyshev bass Pseudo spectral methods are powerful approach for numercal soluton of partal dfferental equatons [34, 35, 36], whch can be traced bac to 970s [37] In pseudo spectral methods [38], there are bascally two steps to obtanng a numercal approxmaton to a soluton of dfferental equaton Frst, an approprate fnte or dscrete representaton of the soluton must be chosen Ths may be done by polynomal nterpolaton of the soluton based on some sutable nodes However, t s well nown that the Lagrange nterpolaton polynomal based on equally spaced ponts does not gve a satsfactory approxmaton to general smooth functons In fact, as the number of collocaton ponts ncreases, nterplant polynomals typcally dverge Ths poor behavor of the polynomal nterpolaton can be avoded for smoothly dfferentable functons by removng the restrcton to equally spaced collocaton ponts Good results are obtaned by relatng the collocaton ponts to the structure of classcal orthogonal polynomals, such as the well-nown Chebyshev-Gauss-Lobatto ponts The second step s to obtan a system of algebrac equatons from dscretzaton of the orgnal equaton In the case of dfferental equatons, ths second step nvolves fndng an approxmaton for the dfferental operator see [37] Many authors have consdered ths technque to solve many problems In [39, 40], pseudo spectral scheme to approxmate the optmal control problems Also, a Legendre pseudo spectral Penalty scheme used for solvng tme doman Maxwells equatons [4] The method of Hermte pseudo spectral scheme s used for Drac equaton [4], and nonlnear partal dfferental equatons [43], respectvely In [44], multdoman pseudo spectral method for nonlnear convecton dffuson equatons was presented onlnear Schrödnger equaton was dscussed n [45] by Tme Space pseudo spectral method wth Chebyshev bass Fnally, [46] pseudo spectral methods used n Quantum and Statstcal Mechancs The organzaton of ths artcle s as follows In Secton, we present some prelmnares about Chebyshev polynomals and drve some tools for dscretzng the ntroduced problem In secton 3, we summarze the applcaton of Chebyshev pseudo spectral method to the soluton of the problem 3 As a result a set of algebrac lnear equatons are formed and a soluton of the consdered problem s dscussed In Secton 4, we present some numercal examples to demonstrate the effectveness of the proposed method To llustrate the valdty and applcablty of the proposed technque, A comparson between the presented method has been obtaned wth cubc B-Splne fnte dfference method n [7, 47] Prelmnares and otatons In ths secton, we gve some notatons about most commonly used set of orthogonal polynomals, Chebyshev polynomals [48, 49] whch are defned on the nterval [-,] and can be determned wth the ad of the followng The Chebyshev polynomals T n x, n=0,,, are the Egenfunctons of the sngular Sturm-Louvlle problem d dx x dt nx dx + n x T nx=0 They are mutually orthogonal wth respect to L nner product on the nterval, wth the weght functon ωx=/ x Ths mply T n xt m xωxdx= d nπ δ nm, where δ nm s the Kronecer delta, d 0 = and d n = n The Chebyshev polynomals satsfy the followng threeterm recurrence relaton T 0 x =, T x=x, T n+ x = xt n x T n x, n, 4 and T 0 x = T x, T x=05t x, T n x = n+ T n+ x n T n x, n 5 The Rodrgues formula for Chebyshev polynomals s obtaned drectly by normalzng approprately; T n x= n n! n x dn { x n! dx n n 05} A unque feature of the Chebyshev polynomals s ther explct relatonshp wth a trgonometrc functon: T n x=cosnarccosx 6 c 06 SP atural Scences Publshng Cor

3 Sohag J Math 3, o, / wwwnaturalspublshngcom/journalsasp 9 In ths wor, we wll use the Chebyshev Gauss Lobatto CGL ponts as nπ x n = cos 7 Let T z denote the Chebyshev polynomal of order, then CGL nodes wll be z 0,,z, as defned n 7 ow, let {φ z} =0 be the Lagrange polynomals based on CGL nodes, that are expressed as [45,50]: φ z= =0, z z z wth the Kronecer property φ z =δ = z, = 0,,, 8 { 0,,, = It s more convenent to consder an alternatve Chrstoffel Darboux formula [3, 45], for = 0,,, where φ z= z c z z T z, 9 {, =0,, c =, Any defned functon f on the nterval [, ] may be approxmated by Lagrange polynomals as fz =0 f φ z, 0 where f ={ fz } =0 Equaton 0 wll be exact when fz s a polynomal of degree at most Equaton 0 can be expressed n the followng matrx form fz Φ F, [ ] where Φ = φ 0 z,,φ z and F=[ fz 0,, fz ]T The frst dervatve to equaton 0 can be expressed as f z =0 f φ z, where φ z s a polynomal of degree, whch can be wrtten as φ z= =0 φ z φ z, =0,, Equaton can be expressed n the followng matrx form: d dz Φ z=φ zd +, 3 where D + s the so called dfferentaton matrx wth dmenson + From the last two equatons,3 we get [D + ], = φ z The entres of the dfferentaton matrx can be defned for CGL ponts cf [5] as the followng c + c z z, +, ==0, [D + ], = 6 z z, =, +, == 6 4 ow, we ntroduce the second order dfferentaton matrx as D + whch s the dervatve to dfferentaton matrx D + The entres to the second order dfferentaton matrx can be defned for CGL ponts cf [5] as the followng [D + ], [D + ], [D z z, +], = [D +],, = =0, 5 Also, any defned functon hx on an arbtrary nterval [a, b] may be approxmated by mang transformaton from z [,] to x [a,b] as: hx =0 hx φ x a, 6 where x = { z + +a} =0 are the shfted CGL nodes assocated wth nterval [a, b] Equaton 6 can be expressed n the followng matrx form: hx Φ xh 7 [a,b] In vew of equatons 3 and 6, we conclude that d dx Φ [a,b] x= Φ [a,b] xd +, 8 For an arbtrary and M, any functon of two varables u :[a,b] [c,d] R may be approxmated by ux,y M =0 =0 where U, = u z U, φ φ M x a y c, 9 d c + +a, d c zm + +c 0 c 06 SP atural Scences Publshng Cor

4 0 G I El Baghdady, M S El Azab: Chebyshev Gauss Lobatto Pseudo Spectral Equaton 9 can be expressed based on Kronecer product n the followng matrx form: ux,y Φ U, [a,b] x ΦM [c,d] y where U s the + M+ vector as the followng form: U=[U 0,0,,U 0,M U,0,,U,M ] T The prevous representatons that are based on Kronecer product, provde some smplfcaton n calculatons when we deal wth our orgnal problem Also by usng frst and second dfferentaton matrces we can approxmate relatve dervatves of any functon from ts expanson as we can see next For example let u be approxmated as n, now we can wrte the frst dervatve to u wth respect to x as the followng: d u x x,y U dx Φ [a,b] x ΦM [c,d] y = Φ [a,b] xd + Φ M [c,d] U y = Φ [a,b] x ΦM [c,d] y D + I M+ U 3 In a smlar way, we can conclude that the frst dervatve to u wth respect to y as the followng: u y x,y Φ d c [a,b] x ΦM [c,d] y I M+ D M+ U 4 3 Chebyshev Pseudo spectral Approxmaton In order to solve problem 3, we approxmate ux, t as: ux,t Φ [a,b] x ΦM [0,T] U, t 5 where the postve and nteger numbers and M are dscretzaton parameters correspondng to space and tme dmensons, respectvely Also we wll consder {x } =0 and {t } M =0 as the CGL nodes correspondng to the ntervals[a, b] and[0, T], respectvely By usng 5 and dfferentaton matrces, we can wrte the dervatves to ux,t as the followng u x x,t Φ [a,b] xd + Φ U, M [0,T] t 6 4 u xx x,t u t x,t T Φ [a,b] xd + Φ M [0,T] t U, 7 Φ [a,b] x ΦM [0,T] td M+ U 8 ow, by substtutng from the prevous equatons n equaton, we obtan [ Φ T [a,b] x ΦM [0,T] td M+ Φ [a,b] xd + Φ M [0,T] t + qx 4px Φ [a,b] xd + ]U= ΦM [0,T] t fx,t 9 ow, for < < and < < M, we collocate the above equaton at the collocaton ponts {x,t }, ote that these collocaton ponts are the nteror ponts not le n ntal or boundary condtons After collocatng, equaton 9 becomes: [ T e + + em+ + D M+ + qx e + + D + e M+ + 4px e + + D + e M+ + ] U = fx,t, =,,, =,,M, 30 where e p s the th row of p p dentty matrx Equaton 30 can be represented n the followng matrx form usng dentty matrx: [ [I] T [I]M+ D M+ 4px [I] D + [I]M+ whch can be formed as + qx [I] D + [I] M+ ] U = F, 3 A U = F, 3 where F and U are the M vectors they tae the followng forms: F = [ f,,, f,m f,,, f,m ] T, U = [U,,,U,M U,,,U,M ] T, and A s a matrx of dmenson M+, that can be defned as [ A = [I] T [I]M+ D M+ + qx [I] D + [I] M+ 4px ] [I] D + [I]M+ For dscretzaton the ntal condton, we substtute 5 n gettng the followng Φ U=u [a,b] x ΦM [0,T] 0 0 x, a x b, ow, for 0<<, we collocate the above equaton at the collocaton ponts{x,0} After collocatng, the prevous equaton becomes: e + + em+ U = u 0 x, 33 c 06 SP atural Scences Publshng Cor

5 Sohag J Math 3, o, / wwwnaturalspublshngcom/journalsasp then n matrx form usng dentty matrx [I] + e M+ U = U 0, 34 whch can be formed as A U = U 0, 35 where U 0 and U are the + vectors, they can be descrbed as the followng forms: U 0 = [u 0 x 0,,u 0 x ] T, U = [U 0,0,,U,0 ] T, and A s a matrx of dmenson + +, that has the followng form A = [I] + e M+ Fnally, to dscrete the boundary condtons, we substtute 9 n 3 Frst, we deal wth the left boundary to fnd the reduced form, then dong the same wth the rght boundary Equaton 3 wll be Φ U=g [a,b] a ΦM [0,T] t t, 36 ow, for < < M, we collocate the above equaton at the collocaton ponts {a,t } for the frst boundary condton After collocatng, the prevous equaton becomes: e + e M+ + U 3 = g t, 37 then n matrx form usng dentty matrx e + [I] M+ U 3 = G, 38 whch can be formed as A 3 U 3 = G, 39 where G and U 3 are the M vectors, they can be descrbed as the followng forms: G = [g t,,g t M ] T, U 3 = [U 0,,,U 0,M ] T, and A 3 s a matrx of dmenson M M+, that has the followng form A 3 = e + [I] M+ Smlarly, we can wrte the equaton of the second boundary condton as the followng form e + + [I]M+ U 4 = G, 40 whch can be formed as A 4 U 4 = G, 4 where G and U 4 are the M vectors, they can be descrbed as the followng forms: G = [g t,,g t M ] T, U 4 = [U,,,U,M ] T, and A 4 s a matrx of dmenson M M+, that has the followng form A 4 = e + + [I]M+ The resultng system of equatons can be descrbed, from collectng equatons 3, 35, 39 and 4, as the followng AU=F, 4 where A s a matrx of dmenson + M+, that has the form A = [A A A 3 A 4 ] For U and F, each one s a vector wth dmensonm+, and tae the followng form U = [U U U 3 U 4 ] T, F = [F U 0 G G ] T After solvng the lnear system descrbed n 4, we can fnd the approxmated soluton to our problem 4 umercal Examples In order to test the utlty of the proposed method, we apply the new scheme to the followng examples whose exact solutons are provded n each case For both examples, we tae = M and to show the effcency of the presented method for our problems n comparson wth the exact soluton Also, to study the convergence behavor of the presented method, we appled the followng laws for dfferent values of and for t = T : The E error norm of the soluton s defned by E = Ux,t ux,t = max U,M ux,t M, The E error norm of the soluton s defned by E = Ux,t ux,t = [ ] / U,M ux,t M, = The condton number K g A of the coeffcent matrx A s gven by K g A= A g A g, g=, All the computatons are carred out n double precson arthmetc usng Matlab 790 R009b To obtan suffcent accurate calculatons, varable arthmetc precson vpa s employed wth dgt beng assgned to be 3 The code was executed on a second generaton Intel Core 540M, 3 Ghz Laptop c 06 SP atural Scences Publshng Cor

6 G I El Baghdady, M S El Azab: Chebyshev Gauss Lobatto Pseudo Spectral Example [5] Consder the problem 3 wth the ntal condton ux,0 = snπx, 0 x, and the boundary condtons are gven as { u0,t=0, 0 t, u,t=0, and the exact soluton ux,t = snπxe πt, n ths case the forcng functon wll be [ fx,t=e π t π snπx px ] + qxπ cosπx a Exact soluton Table : E error, E error, condton number of g=, g = wth dfferent values of for Example E E K A K A CPUs 6 55E E e+0 39e E-06 3E-05 e+03 7e E E-07 5e+03 39e E-09 75E-08 07e+04 75e E-0 48E-0 97e+04 4e E- 03E- 337e+04 57e E-4 9E-3 54e+04 43e E-4 9E-3 87e e In Table, we tae px = x/ + x and qx = e x In Comparng wth cubc B-Splne fnte dfference method [7, 47], the maxmum error was 444E 05 at T = for x = 00 and t = 000, mang CPU-tme equal to sec Example [5, 6] Consder the problem 3 wth the ntal condton ux,0=e 5x cos π x+05snπ x, 0 x, and the boundary condtons gven by { u0,t=e C 0 t, u,t=05e 5 C0t 0 t,, and the exact soluton ux,t=e 5x C 0t cos π x+05snπ x, n ths case the forcng functon wll be fx,t = {cos π x[ C 0 +C qx px5c + π C ] where + sn π x[ C 0 4 +C qx px5c π C ]} e 5x C 0t, C 0 = π , C = 5+ π 8, C = 5 4 π b umercal soluton Fg : Exact and umercal solutons for px, qx wth x [0,] and t [0,] at = 0 for Example Table : E error, E error, condton number of g=, g = wth dfferent values of for Example E E K A K A CPUs 6 06E-03 35E-03 36e+0 70e E-05 3E e+0 57e E-07 4E-07 5e+03 34e E-09 4E e+03 9e E- 388E- 780e e E- 76E- 3e+04 0e E- 90E- 0e+04 7e In Table, we tae px = xe x / + x and qx = e x /+x In Comparng wth cubc B-Splne fnte dfference method [7, 47], the maxmum error was 45977E 03 at T = for x=00 and t = 000, mang CPU-tme equal to 57 sec c 06 SP atural Scences Publshng Cor

7 Sohag J Math 3, o, / wwwnaturalspublshngcom/journalsasp 3 a Exact soluton b umercal soluton Fg : Exact and umercal solutons for px, qx wth x [0,] and t [0,] at = 8 for Example 5 Concluson In ths wor, we apply Chebyshev Pseudo spectral method for one-dmensonal advecton dffuson equaton wth Vrable coeffcents on Chebyshev Gauss Lobatto nodes The man advantage of usng Chebyshev scheme nstead of usng Legendre one s that ts quadrature ponts have explct and smple expressons The dfferentaton matrces are used to represent the unnown functons Two examples are ntroduced n ths artcle to show that the proposed numercal procedure s effcent and provdes very accurate results even wth usng a small number of collocaton ponts The Pseudo spectral scheme s a powerful approach for the numercal soluton of parabolc advecton dffuson equaton References [] B J oye, umercal Solutons of Partal Dfferental Equatons, Elsever Scence Ltd, Unted Kngdom, 98 [] B J oye, umercal Soluton of Partal Dfferental Equatons, Lecture otes, 990 [3] J Isenberg and C Gutfnger, Heat transfer to a dranng flm, Int J Heat Transf 6, pp [4] J Y Parlarge, Water transport n sols, Ann Rev Fluds Mech, [5] Q Fattah and J A Hoopes, Dsperson n ansotropc homogeneous porous meda, J Hydraul Eng, pp [6] P C Chatwn and C M Allen, Mathematcal models of dsperson n rvers and estuares, Ann Rev Flud Mech 7, pp [7] F M Holly and J M Usseglo Polatera, Dsperson smulaton n two dmensonal tdal flow, J Hydraul Eng, pp [8] J R Salmon, J A Lggett and R H Gallager, Dsperson analyss n homogeneous laes, Int J umer Meth Eng 5, pp [9] L Lapdus and R Amundston, Mathematcs of absorpton n beds, J Physcal Chem 56 8, pp [0] Z Zlatev, R Berowcz and L P Prahm, Implementaton of a varable stepsze varable formula n the tme-ntegraton part of a code for treatment of long-range transport of ar pollutants, J Comput Phys 55, pp [] C R Gane and P L Stephenson, An explct numercal method for solvng transent combned heat conducton and convecton problems, Int J umer Meth Eng 4, pp [] M H Chaudhry, D E Cass and J E Ednger, Modellng of unsteady flow water temperatures, J Hydraul Eng 09 5, pp [3] Kumar, Unsteady flow aganst dsperson n fnte porous meda, J Hydrol 63, pp [4] V Guvanasen and R E Voler, umercal solutons for solute transport n unconfned aqufers, Int J umer Meth Fluds 3, pp [5] Joan Goh, Ahmad Abd Mad and Ahmad Izan Md Ismal, Cubc B Splne Collocaton Method for One-Dmensonal Heat and Advecton Dffuson Equatons, Journal of Appled Mathematcs Hndaw Publshng Corporaton Vol 0, pp 8 0 [6] A Mohebb and M Dehghan, Hgh order compact soluton of the one-dmensonal heat and advecton dffuson equatons, Appled Mathematcal Modellng Vol 34 0, pp [7] W K Zahra, W A Ouf and M S El-Azab, Cubc B- splne collocaton algorthm for the numercal soluton of newell wghtehead segel type equatons, Electronc Journal of Mathematcal Analyss and Applcatons Vol -, pp [8] W M Abd-Elhameed, E H Doha and Y H Youssr, Effcent spectral-petrov-galern methods for thrd- and ffth-order Jacob polynomals, Quaestones Mathematcae, 36, pp 5 38, 03 [9] E H Doha, A H Bhrawy, M A Abdelawy and R M Hafez, A Jacob collocaton approxmaton for nonlnear coupled vscous Burgers equaton, Central European Journal of Physcs,, pp, 04 [0] E H Doha, A H Bhrawy, M A Abdelawy and R A Van Gorder, Jacob-Gauss-Lobatto collocaton method for the numercal soluton of + nonlnear Schrödnger equatons, Journal of Computatonal Physcs, 6, pp 44 55, 04 [] E H Doha, A H Bhrawy, D Baleanu and R M Hafez, A new Jacob ratonal-gauss collocaton method for numercal soluton of generalzed Pantograph equatons, Appled umercal Mathematcs, 77, pp 43 54, 04 c 06 SP atural Scences Publshng Cor

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