NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS

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1 IJRRAS 8 (3 September NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc Scences Faculty For Grls, Kng Abdul Azz Unversty, Jeddah, K.S.A. Emal: h.o.bak@hotmal.com ABSTRACT A sutable spatal dscretsaton wth 7 pont formula n the method of lnes s ntroduced. It s shown that, ths method s capable of solvng some knds of Burgers' equaton and comparsons are made wth publshed analytc solutons. 1. INTRODUCTION The method of lnes s a general technque for solvng partal dfferental equaton (PDE's by typcally usng fnte dfference relatonshps for the spatal dervatves and ordnary dfferental equatons for the tme dervatve, as dscussed by schesser[1]. The method of lnes (MOL semdscretzaton approach whch used to transform the model (PDE nto a system of frst order lnear ordnary dfferental equatons (ODE's. The partal dervatve wth respect to the space varable s approxmate by a second order fnte dfference approxmaton used 3-pont central dfference and some tme ths requres a specfcaton of the dependent varable values outsde the endponts. In [] Hcks and We proposed that, explct consderaton of ths problem can be avod by the use of non-central dfference approxmatons for example, wth 5-pont dfference approxmaton, and they proofed that the egenvalues of the 5-pont non-central dfference method of lnes matrx are real and negatve whch assures stablty. In[3]Javd presented new method by combnaton of(moland matrx free modfed extended backward dfferental formula. In ths paper, numercal schemes are developed for obtanng approxmate solutons to Burger's equaton n three dfferent cases wth a non-central 7-pont formula then the solutons are compared usng the results of numercal experments wth 3and 5- ponts formulas. In secton stablty analyss for these formulas was presented, ths allows the man contrbuton of the paper to be gven n secton 3, where ths approxmaton was used n solvng Burger's equaton for arbtrary ntal condtons, we have appled successfully 7-pont formula n the method of lnes. Fnally, n secton 4 some conclusons and further developments are presented.. BURGERS' EQUATION AND MOL Consder Burgers' equaton n the form u t u u x u where s postve parameter and homogeneous boundary condtons are used. A study of the propertes of Burger's equaton s of great mportance due to the equaton's applcatons n the approxmate theory of flow through a shock wave travelng n a vscous flud and the Burger's model turbulence. Burger's equaton s a very good example for several reasons( as Byrne and Hndmarsh mentoned n [4] : It s nonlnear. The exact soluton of the PDE s known. It can be thought of as hyperbolc problem wth artfcal dffuson for small. It s some tmes used n boundary layer calculaton for the flow of vscous flud. It s very nearly a standard test problem for PDE solvers. Ths equaton recent solved by dfferent methods such as n [5,6,7]. Apply the method of lnes to equaton (.1 we dscretzed the partal dervatves wth respect to the space varables to result n approxmatng system of ODEs n the varable t. and n the MOL approach, the ODEs are ntegrated drectly wth a standard code for the task. The smplest method of spatal dscretzaton s to dscretze along the x axs wth a unform mesh and to replace all spatal dervatves n(4.1 by fnte dfference : Let 1 N 1 and u ( u(, 0,1,,..., N 1.The resultng s a system of ODEs, for the method of lnes approach to solvng (.1. xx (.1 38

2 IJRRAS 8 (3 September 011 Bakodah Parabolc and Burgers' Equatons 3. STABILITY ANALYSIS The stablty analyss of the method of lnes(mol for partal dfferental equatons represents the most mportant factor for ther solutons and at the same tme s a crtcal factor that should be handled carefully the mportance les n ts unque ablty of judgng acceptable soluton for the gven equaton, of beng crtcal s due to ts dependence on the nature of the egenvalues of the matrx representaton connecton wth ther number. The stablty analyss[8] consttute the essental study of the numercal soluton of partal dfferental equatons n general ths s because such study provdes the means by whch the step sze and the numercal ntegraton scheme for the gven dfferental equaton could be selected so as to secure manageable numercal soluton. Regardng the method of lnes for parabolc partal dfferental equaton (n two varables, t can be classfed accordng to the nature of the resultng system n connecton wth the drecton of dascretzaton as shown n the followng table : Table (1 Nature of system n connecton wth the drecton of dscretzaton Dscretzaton drecton x - drecton t - drecton Nature of the resultng system Intal value type n ODE. Boundary value type n ODE. The stablty analyss of the method of lnes for dscretzaton that produce ntal value type n ordnary dfferental equatons s well studed by varous authors and n vew of the too complcated behavor of the boundary value problems than that of the ntal value one,t s not surprsng, that no serous attempts have been made to analyze the stablty of the method of lnes for dscretzaton that produce boundary valued problems. In order to analyze the stablty of Burger's equaton Frst requres that we replace the spatal dervatve, u x xx u t u xx u u x (3.1 u, wth an algebrac approxmaton. If we choose a 7- pont dfference approxmaton for frst dervatve and second dervatve,[9], then substtuton of these dervatves n equaton (3.1 gves the ordnary dfferental equatons that must be ntegrated numercally at the spatal grd ponts, =,,N-1. du dt u 8u 3 108u 1080u 1960u 1080u 108u 8u 1u 108u 540u 0u 540u 108u 1u / 70h ( / 70h We can now nqure about the egenvalue of the N ordnary dfferental equatons of equaton (3.. Thus,we assume a tral soluton and substtute t nto equaton (.. However, the tral soluton must take nto account the varaton of u ( x, wth both x and t or and t. Therefore, we assume, as n [1], a product soluton : u( x, C ( ( x Further, n accordance wth a method proposed by Von Neumann, we assume the x dependency (x the form : jkx ( x e, j 1 Where k s a Fourer number. Substtuton of equatons (3.3 and (3.4 n equaton (3. gves : d C e dt 8e jk ( x 3x C 70 h 1e jkx 108e max u jk ( x 3 x C 70h 108e Where. jk ( x x jk ( x x 1080e 540e jk ( x x jk ( x x 1960e jkx 540e 1080e jk ( x x jk ( x x 108e 108e jk ( x x jk ( x x 1e 8e jk ( x 3 x (3.3, to be of jk ( x 3x (3.4 39

3 IJRRAS 8 (3 September 011 Bakodah Parabolc and Burgers' Equatons d jkx e 16cos(3kx 16cos(kx 160cos( kx 1960 dt 70h j 4sn(3kx 16sn(kx 1080sn( kx 70 h Note that : -1960= , so : d jkx e dt 70h ( cos(3kx (16 16 cos(kx ( cos( kx j 4sn(3kx 16sn(kx 1080sn( kx 70 h d e dt jkx j 70 h 70h Thus, ( 16sn (3kx 16sn (kx 160sn ( kx 4sn(3kx 16sn(kx 1080sn( kx (3.5 Equaton (3.5 wrtten n terms of an egenvalue becomes : Where : d dt ( 16sn (3kx 16sn (kx 160sn 70h j 70 h 4sn(3kx 16sn(kx 1080sn( kx (3.6 ( kx Note that all N egenvalues gven by equaton (3.6 have negatve real parts, whch mean that the system of N frst order ordnary dfferental equatons are stable. 4. The test problems : To measure the accuracy of the numercal method of lnes we compute the dfference between the analytc and numercal solutons at each meeash pont after specfed tme steps, and use these to calculate the dscrete L - and L - error norms. (a Frst test problem [10]: Frst we take the ntal condton and Drchlet boundary condtons drectly from an exact soluton: u( x, 1/(1 exp( x / t / 4 (4. where 1, 0 x 1, t 0. Hence we fnd that : Wth 3-pont formula : ( The effect of ncreasng node ponts (nx t 0.001, t 0.1 nx Norm ( ( ( ( (-6 Table(11 Normnf ( ( ( ( (-5 330

4 IJRRAS 8 (3 September 011 Bakodah Parabolc and Burgers' Equatons ( The effect of ncreasng tme : tme ( The effect of change t t : t 0.001, nx 51 Norm ( ( ( ( ( ( ( ( ( (-5 Table(1 t 0.1, nx 51 Norm Normnf ( ( ( ( ( ( ( ( ( (-4 Normnf ( ( ( ( ( ( ( (-4 Table(13 second we compare use of 3-pont formula, 5-pont formula and 7-pont formula results: t 0.5, nx 51, t 0.01 N pont formula Norm Normnf 3 pont formula.0337( (-5 5 pont formula 3.031( (-4 7 pont formula ( (-5 Table(14 It can be seen that the accuracy of the numercal solutons wth 7-pont formula s more effcent. but when t 0.1 then,7-pont formula wll be the best also: t 0.1, nx 11, t 0.01 N pont formula 3 pont formula 5 pont formula 7 pont formula Norm ( ( (-6 Table(15 Normnf 5.575( ( (-5 Fgures(7,(8 and(9 show us the behavor of the numercal and analytcal soluton for the three cases : nx 51, t 0.01, t 0.5 : 331

5 IJRRAS 8 (3 September 011 Bakodah Parabolc and Burgers' Equatons fgure(7 fgure(8 (b second test problem [11]: A second analytc soluton s : fgure(9 1 t u( x, 0.5{1 tanh[ ( x 15]} 4 0 x wth boundary condtons : we take as ntal condton (4.3 at t=0 over the rang 30 u ( 0, u(30, 0 And the numercal solutons wth 7-pont formula also s the better n ths case as show from table (16: t 1, nx 11, t N pont formula Norm Normnf 3 pont formula pont formula pont formula Table(16 Fgures(10,(11 and(1 show us the behavor of the numercal and analytcal soluton for the three cases : (4.3 33

6 IJRRAS 8 (3 September 011 Bakodah Parabolc and Burgers' Equatons fgure(10 fgure(11 fgure(1 (c thrd test problem [10]: As our last test example, consder the analytc soluton: u( x, ( x / /(1 exp( x / 4 (4.4 where t 0 exp(1/ 8. we take as ntal condton, (4.4 at t=1 over the rang 0 x 1wth boundary condtons, u ( 0, u(1, 0. The numercal results show us the hgh dfference between 3,5 and 7 pont formula, whch seen that 7-pont formula have better accuracy than 3 and 5 formula n solvng Burger's equaton. The last results n table (17,(18,(19,(0 and (1. Fgures (13,(14, (15,(16, (17 and (18 show us the behavor of the numercal and analytcal soluton for the 3,5 and 7 pont formula: N pont formula 3 pont formula 5 pont formula 7 pont formula t, nx 1, t Norm 8.504( (-4 Table(17 Normnf

7 IJRRAS 8 (3 September 011 Bakodah Parabolc and Burgers' Equatons N pont formula 3 pont formula 5 pont formula 7 pont formula t 3, nx 11, t Norm (-4 Table(18 Normnf nx ( The effect of ncreasng node ponts (nx n 7-pont forula: t 0.001, t 1.1 Norm 1.318( ( ( (-5 Table(19 Normnf ( The effect of ncreasng tme : t 0.001, nx 1 tme Norm Normnf ( ( ( (-4.037(-4 Table( t ( The effect of change t : t 1.1, nx 11 Norm ( (-5 Table(1 Normnf The accuracy of the numercal solutons wth 7-pont formula could be seen also from table (. nx 51, t tme Norm Normnf ( ( ( ( ( ( (-4 Table( 334

8 IJRRAS 8 (3 September 011 Bakodah Parabolc and Burgers' Equatons 1. when nx 1, t 1.7, t fgure(13 fgure(14 fgure(15. when nx 1, t, t fgure(16 335

9 IJRRAS 8 (3 September 011 Bakodah Parabolc and Burgers' Equatons fgure(17 fgure(18 4. CONCLUSIONS AND FURTHER DEVELOPMENTS Our am s to develop a good formula wth sutable number of ponts wth hgh accuracy for the numercal soluton of parabolc equatons usng the method of lnes. Hccks and We haven't prove stable for dfference approxmatons of order greater than 5-pont manly because, as they mentoned, for a number of problems, use of 7,9,11 pont non- central dfference approxmatons dd not effect any sgnfcant mprovement n convergence tme over use of the 5-pont formulaton. But n ths paper we fnd that usse 7-pont formula effect n sgnfcant mprovement for a number of examples, and n recent one can tray to use 9 or 11 pont non- central dfference approxmatons, as specally as, wth that problems whch have less accuracy wth 3-pont formula. REFERENCES [1]. Schesser, W. E.," The Numercal Methods of Lnes Integraton of Partal Dfferental Equatons ", Acadmc Press, San Dego,(1991. []. Hcks, J. S. and J. We, " Numercal Soluton of Parabolc Partal Dfferental Equatons wth two-pont Boundary Condtons by use Method of Lnes ", J.Acm,Vol.14,No.3(1967, [3]. Javd,M.,"A Numercal Soluton of Burger's equaton based on modfed extended BDF scheme ",Int. Math.Forum, 1,No.3( [4]. Byrne, G.D. and A.C. Hndmarsh, " Stf ODE Solvers: A revew of Current and Comng Attractons ", Journal of Computatonal physcs,70,(1987,1-6. [5]. Mamaloukas,C. and Spartals,S.,"Decomposton method n Comparson wth Numercal Solutons of Burger's equaton",rato Mathematca,18( [6]. Knapp,R.,"A Method of Lnes Framework n Mathematca", (JNAIAM,vol.3,no.1-( [7]. Mamaloukas,C., Haldar,K.and Mazumdar, H.P."Applcaton of double decomposton to pulsatle flow, J. of Computatonal &Appled Mathematcs 10, Issue1-( [8]. BAKODAH, H.O. and SHARAF,A.A.," On Stablty of the method of Lnes", To appear n Orental, 011. [9]. Journal of Mathematcs. [10]. Sharaf, Amr A. and Bakodah H. O.,"A good Spatal Dscretzaton n the Method of Lnes", Appled Mathematcs and computaton, Vol. 171-, pp (005. [11]. Al,A.H.A.,G.A.Gardner and L.R.T.Gardner,"A Collocaton Soluton for Burger's equaton usng Cubc B- Splne Fnte Elements", Computer Methods n Appled mechancs and Engneerng 100(199, [1]. Gardner, L.R.T., G.A.Gardner and, A.H.A.Al "A Method of Lnes Soluton for Burger's equaton ", Computer Methods n Appled mechancs and Engneerng December(1991,