International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September ISSN

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1 Iteratoal Joral o Scetc & Egeerg Research, Volme 5, Isse 9, September-4 5 ISSN Nmercal Implemetato o BD va Method o Les or Tme Depedet Nolear Brgers Eqato VjthaMkda, Ashsh Awasth Departmet o Mathematcs, NIT Calct, Ida-676. Abstract I ths paper a ecet codtoally stable mercal scheme s proposed or solvg oe dmesoal qas lear Brgers eqato. The proposed scheme comprses o sem dscretzato va method o les or the space varable ad backward deretato ormla o order two (BD or the tme varable. The method o les redces the qas lear partal deretal eqato to olear ordary deretal eqatos at each ode pot. The resltg olear system s solved by a ecet st solver kow as BD. BD s a mplct solver whch leads to olear algebrac system ad the resltg olear algebrac system s learzed va Taylor seres. Ths learzato techqe s easy to mplemet ad the accracy o the method wll rema chaged. The learzed system o algebrac eqatos s solved sg MATLAB 8.. The proposed scheme s mplemeted o test examples ad t has bee observed that the mercal solto les very close to the exact solto. Varos mercal expermets have bee carred ot to demostrate the perormace o the method. Idex Terms Brgers eqato ; Kematc vscosty; Method o les; Backward Deretato ormla; Taylor seres. INTRODUCTION solto or ths eqato does ot exst. Hece several scetsts ad mathematcas are I ths paper, we cosder the qaslear oe terested dg ts mercal solto. So dmesoal Brgers eqato ar, varos mercal methods have bee developed sch as, te elemet method t + x vxx, x [,] adt [, T ] method [5], Adoma's decomposto method (.a [], Petro-Galerk method [], explct ad exact-explct te derece method [], a wth tal codto mxed te derece ad bodary elemet ( x, ( x, x, (.b approach [], B- sple te elemet method [4], Crak-Ncolso scheme [], Doglas te derece scheme [6], meshless method o les ad bodary codtos [9], psedospectral method ad Darvsh's (, t, t T, (.c (, t, t T. (.d precodtog [7], lattce Boltzma method [8] ad Haar wavelet qaslearzato approach []. I ths paper, qas lear oe-dmesoal Brgers' where v > s the kematc vscosty parameter eqato s solved by Method o les (MOL ad ( x s gve scetly smooth cto. whch the spatal dervatves are approxmated by te dereces. The qas lear partal Ths eqato s kow as Brgers eqato deretal eqato gets coverted to a olear whch s amed ater J. M. Brgers [5], [6] de to system o ordary deretal eqatos tme hs eormos cotrbtos. It was trodced varable. Ths system s solved by Backward by Batema [] 95. The olear physcal Deretato ormla o order two (BD- pheomea trblece s modelled by ths combed wth Taylor seres expaso. Taylor eqato. It s strctre s smlar to Naverseres expaso s sed or learzato ad Stoke s eqato ad hece precse aalytc 4

2 Iteratoal Joral o Scetc & Egeerg Research, Volme 5, Isse 9, September-4 54 ISSN lear algebrac eqatos are solved drectly thereby creasg the ececy o the proposed scheme. The proposed method has accracy o order two space ad tme. (,, N, t + ( λ λ ( λ λ + λ, (. where,,... N Nmercal Scheme The Nmercal scheme proposed ths paper comprses o Method o Les (MOL, Backward Deretato ormla o order two (BD ad Learzato techqe. MOL s a semdscretzato techqe whch dscretzato s doe oly alog the spatal drecto. We dvde the spatal drecto to N + eqally spaced pots wth space terval x N. Spatal dervatves are approxmated sg cetral derece scheme as gve below. x x ( x, t ( t ( t + ( x, t +,,, N. 4 U U U + t U, t,,... x (.4 ( t ( t + ( t the solto at rst tme level.e. U s obtaed,,, N. rom BD-. ( x Backward Deretato ormla o order oe (BD- U U + ( t ( U, t,,,... M U s the tal codto ad U ( [ (,... - ( ]. Sbstttg Brgers eqato Eq. (., ad takg to accot that (t ad N(t we obta a system o olear ordary deretal eqatos wth tal codto d dt ( t v h ( ( t ( t + ( t h ( t ( ( t ( t + + ( ( x,,, N where, (t (x, t, ths system o (N- deretal eqatos ca be wrtte matrx orm as du ( U, (. dt U ( U T where, U ( t [ ( t, N ( t ]., s a olear cto o U wth elemets j gve as ollows. λ v, λ ( x ( x The system (. s a olear system o ordary deretal eqatos whch ca be solved by tegratg tme varable. Dvde the tme terval to M+ eqally spaced pots wth tme step t M. or tme tegrato we se Backward Deretato ormla o order two gve below... Backward Deretato ormla o order two (BD- ( ( ( M Sce the system (. s olear, t reqre solvg a olear algebrac eqato at each tme level. Ths ca be avoded by sg the learzato techqe. Learze by Taylor seres ( ( ( ( U ( U + J ( U U + ( t, (.5 where ( ( ( N ( J ( ( ( N N N N s the Jacoba matrx at the th tme level. Sbstttg Eq. (.5 Eq. (.4 we get, U ( ( ( t 4 U U ( ( ( ( [ ( U + J ( U U ] + 4

3 Iteratoal Joral o Scetc & Egeerg Research, Volme 5, Isse 9, September-4 55 ISSN t ( ( 4 t ( ( I J U I J U + ( ( t U ( U C exp [ cos( πx ] ( cos πx dx, πv (.8c ( t ( 4 t ( ( U I J I J U + I where t ( J ( ( U t ( ( I J U (.6 ( J s the Jacoba matrx at the th tme level. Hece the above scheme s learzed. Ulke Newto s method at each tme step we eed oly to solve lear algebrac eqatos Eq. (.6 whch take less comptato tme..numerical RESULTS AND DISCUSSION Several test expermets were carred ot to show the ececy ad adaptablty o the proposed mercal scheme. We have compared the compted solto wth exact solto or TABLE deret vales o kematc vscosty v ad at deret vales o al tme. Test Problem Cosder the Brgers Eqato x T Compted t + x vxx, x [,] adt [, T ] (.7a Solto wth the tal codto ( x, s( x, x, π (.7b ad the homogeeos bodary codtos (, t (, t, t T. (.7c The exact solto o the problem s ( ( ( C exp π vt s πx x, t πv C + C exp( π vt cos( πx (.8a where, C exp [ cos( πx ] (.8b πv dx, TABLE Comparso o the mercal solto wth the exact solto at deret space pots or test problem at T. or v. ad t.. x Compted Solto Exact Solto Ν Ν 4 Ν Comparso o the mercal solto wth the exact solto at deret space pots or test problem at., x.8 ad t.. v Exact Solto

. Physcal behavor o mercal soltos o test problem at deret tmes D or

. Nmercal solto o test problem or x.5 ad deret vales o v ad t, (a v.

4 Iteratoal Joral o Scetc & Egeerg Research, Volme 5, Isse 9, September-4 56 ISSN g a g b g b g.. Physcal behavor o mercal soltos o test problem at deret tmes D or x.5, (a v.5, t., T. (b v.5, t., T.. g. a g.. Nmercal solto o test problem or x.5 ad deret vales o v ad t, (a v., t., T. (b v., t., T.. g a g. b g.. Physcal behavor o mercal soltos o test problem cotor plot or x.5, (a v., t.,t (b v., t.,t. 4

5 Iteratoal Joral o Scetc & Egeerg Research, Volme 5, Isse 9, September-4 57 ISSN The Brgers eqato has bee solved both aalytcally ad mercally by the scheme proposed ths paper. Table shows comparso betwee compted ad exact solto wth deret mber o parttos o X-axs or kematc vscosty, v.. It s clear that as mber o partto rees, the approxmate solto les closer to exact, dcatg the cosstecy o the proposed scheme. I Table, exact ad compted soltos are compared at deret tme levels T.,, or kematc vscosty, v.. gre shows that mercal solto agrees exactly wth aalytc solto at each odal pots. The physcal behavor o compted solto s depcted gs. ad throgh cotor ad srace plots or deret vales o kematc vscosty v.,.5,.,.5. 4 CONCLUSION I ths paper, Brgers eqato has bee solved by sem dscretzato techqe ad backward deretato ormla o order two (BD. Ths scheme s tested o test example ad mercal solto have bee compared wth exact at deret tmes, or modest vales o kematc vscosty. The mercal reslts shows excellet agreemet wth exact whch shows the accracy o the proposed scheme. Learzato techqe redces the comptatoal tme as well as cost makg the preset mercal scheme ecet tha the schemes lteratre. REERENCES [] S. Abbasbady, M. T. Darvsh, A mercal solto o Brgers eqato by moded Adoma method, Appl. Math. ad Compt. 6 ( [] H. Batema, Some recet researches moto o lds, Mo. Weather Rev. 4, ( [] A.R. Bahadr, Mstaa Saglam, A mxed te derece ad bodary elemet approach to oe-dmesoal Brgers eqato, Appl. Math. ad Compt., 6 ( [4] E. R. Beto ad G. W. Platzma, A table o soltos o the oe dmesoal Brgers Eqato, Qart. Appl. Math. ( [5] J. M. Brgers, A mathematcal model llstratg the theory o trblece, Adv. Appl. Mech. ( [6] J. M. Brgers, Mathematcal examples llstratg relato occrrg the theory o trblet ld moto, Tras. Roy. Neth. Acad. Sc Amsterdam 7 (99-5. [7] M. T. Darvsh ad M. Javd, A mercal solto o Brgers eqato by psedospectral method ad Darvsh s precodtog, Appl. Math. ad Compt., 7 ( [8] Y Gao, L-Ha Le, Bao-Chag Sh, Nmercal solto o Brgers eqato by lattce Boltzma method, Appl. Math. ad Compt. 9 ( [9] Srajl Haq, Arshad Hssa, Marja Udd, O the mercal solto o olear Brgers type eqatos sg meshless method o les, Appl. Math. ad Compt. 8 ( [] B. M. Herbst, S. W. Schoombe, D. Grths, A. R Mtchell, Geeralzed Petro- Galerk Method or the mercal solto o Brgers eqato, It J. Nmer. Methods Eg. ( [] Ram Jwar, A Haar wavelet qaslearzato approach or mercal smlato o Brgers eqato, Compter Physcs Commcatos 8 ( 4-4. [] Moha. K. Kadalbajoo, A. Awasth, A mercal method based o Crak- Ncolso scheme or Brgers eqato, Appl. Math. ad Compt. 8 ( [] S. Ktlay, A. R. Bahadr, A. Ozdes, Nmercal solto o the oe-dmesoal Brgers eqato: explct ad exact explct te derece methods, J. Compt. Appl. Math. ( [4] S. Ktlay, A. Ese, I. Dag, Nmercal soltos o the Brgers eqato by the least-sqares qadratc B-sple te elemetmethod, J. Compt. Appl. Math. 67 (4 -. [5] T. Oz, E. N. Aksa, A. Ozdes, A te elemet approach or solto o Brgers eqato, Appl. Math. ad Compt. 9( [6] K. Padey, L. Verma, A. K. Verma, O a te derece scheme or Brgers eqato, Appl. Math. ad Compt. 5(

u(x, t) = u 0 (x ct). This Riemann invariant u is constant along characteristics λ with x = x 0 +ct (u(x, t) = u 0 (x 0 )):

u(x, t) = u 0 (x ct). This Riemann invariant u is constant along characteristics λ with x = x 0 +ct (u(x, t) = u 0 (x 0 )): x, t, h x The Frst-Order Wave Eqato The frst-order wave advecto eqato s c > 0 t + c x = 0, x, t = 0 = 0x. The solto propagates the tal data 0 to the rght wth speed c: x, t = 0 x ct. Ths Rema varat s costat