POLYNOMIAL BASED DIFFERENTIAL QUADRATURE FOR NUMERICAL SOLUTIONS OF KURAMOTO-SIVASHINSKY EQUATION

Transcription

1 POLYOMIAL BASED DIFFERETIAL QUADRATURE FOR UMERICAL SOLUTIOS OF KURAMOTO-SIVASHISKY EQUATIO Gülsemay YİĞİT 1 and Mustafa BAYRAM *, 1 School of Engneerng and atural Scences, Altınbaş Unversty, Istanbul, TURKEY * Department of Computer Engneerng, Istanbul Gelşm Unversty, Istanbul, TURKEY * Correspondng author; E-mal: mbayram@gelsm.edu.tr In ths study, a numercal dscrete dervatve technque for solutons of Kuramoto-Svashnksy equaton s consdered. Accordng to the procedure, dfferental quadrature algorthm s adapted n space by usng Chebyshev polynomals and explct scheme s constructed to dscretze tme dervatve. Sample problems are presented to support the dea. umercal solutons are compared wth exact solutons and also prevous works. It s observed that the numercal solutons are well matched wth the exact or exstng solutons. Key words: Kuramoto-Svashnsk equaton, The dfferental quadrature, Chebyshev polynomals. 1. Introducton umercal technques to solve partal dfferental equatons whch model many real lfe phenomenon, are wdely used due to ther fast and effectve outcomes. Today, solvng these knds of problems both analytcally or numercally attract many scentst. In ths content Dfferental Quadrature method s consdered to solve Kuramoto-Svasnsky equaton. The Kuramato-Svashnky equaton has been presented as models of phase turbulence n reactondffuson systems [1, ], plasma nstabltes and flame front propagaton [3]. The model equaton has been wdely studed analytcally and numercally. Collocaton methods based on Chebyshev spectral scheme [4], quntc B-splnes [5], exponental cubc B-splnes [6], have been consdered. Solutons of KS equaton analyzed by usng varous methods such as fnte dfference methods [7], dscontnuous Galerkn method [8], numerc meshless method for space dervatves usng radal bass functon [9], He s varatonal teraton method [10]. Rademacher and Wattenberg studed on vscous shocks for the model equaton [11]. Also, some control results of the equaton are presented [1, 13]. Dfferental Quadrature s dscrete dervatve method to solve ordnary or partal dfferental equatons whch gves numercal results effectvely. The method presented by Bellman and hs assocates [14,15]. The dea was to gve a new perspectve for prevous numercal technques n solvng problems. Snce then, the way has been adapted n a wde range of applcatons. As to the the dea, the dervatve of a functon s defned as a weghted lnear sum of the functon values at all grd ponts related to the used drecton. So, the term weghtng coeffcents occurs, and to obtan these coeffcents, generally polynomals are chosen as test functons whch can be obtaned by polynomal approxmaton theory.

2 In the begnnng, Bellman proposed the dea of two polynomal-based methods for computaton of the weghtng coeffcents for the frst order dervatve. Power functon was used as test functon for the frst approach. And the second one he choose test functon as Legendre polynomals [14, 15]. Shortly after, many new dfferent approaches have been presented wth applcatons to many dfferent knd of engneerng problems. Popular ways have been suggested such as polynomals, Splne functons or Fourer seres expanson. Quan and Chang [16, 17] used Legendre nterpolaton polynomals as test functons then obtaned explct formulaton to fnd the weghtng coeffcents. Shu [18], gave a powerful way as a combnaton Bellman s and Quang-Chang s approaches. Also, frst and hgher order dervatve formulatons were analyzed n detal based on polynomal approaches and Fourer seres expanson approaches usng dfferent knd of grd ponts. Stablty analyss based on egenvalue dstrbuton was explaned together wth dfferent tme ntegraton schemes. Shu also presented the relatonshps between fnte dfference and collocaton methods wth the dfferental quadrature method [18]. The DQ method has been effectvely used n areas such as materal scence, thermal and structural mechancal analyss, physcs and bology. And t can be seen that ths technque gves accurate solutons wth tme savng computatons [19]. Cvan and Splepcevch appled ths method to both Posson equaton [0], and to mult-dmensonal problems [1]. Saka et al. consdered equal wdth equaton (EW) by usng three methods ncludng cosne expanson based dfferental quadrature []. Alper et al, studed on a wde of range of problems usng Splne functons or polynomals. For tme dscretzaton they used fourth order Runge-Kutta scheme and stablty analyss s examned [3, 4, 5]. Sar and Güraslan nvestgated the polynomal based method for generalzed Burgers-Huxley equaton together thrd order thrd-order Runge-Kutta scheme for temporal dscretzaton, wthout usng lnearzaton [6]. Mttal et al, used Bernsten polynomals to acqure the weghtng coeffcents [7]. In our work, we consder egenvalue dstrubton to check the stablty, also, several theoretcal works have been establshed for stablty analyss of nonlnear partal dfferental equatons [8,9]. The Kuramoto-Svashnsky (KS) equaton s a nonlnear partal dfferental equaton gven by, 4 u u u u u x x x t T t x x x 0, [ 4 0, ], (0, ] (1) along wth the boundary condtons u( x, t) g, u( x, t) g, u ( x, t) 0, u ( x, t) 0, x 0 x u ( x, t) 0, u ( x, t) 0, xx 0 xx () and ntal condton u( x,0) u (3) where represents growth of the lnear stablty and shows surface tenson. When 0, the term surface tenson s removed from the equaton, then the equaton becomes Burgers' equaton [5]. In ths work, t s used Chebyshev polynomal approxmatons to obtan numerc solutons. When the method s appled for the dervatves, dfferental equaton s reduced to lnear system of equaton, wth the mplementaton of boundary or ntal condtons, matrx equaton can be solved to obtan the desred soluton. 0

3 . The Dfferental Quadrature Method Consder a suffcently smooth functon f() x on a closed nterval [ ab, ]. Dervatve of the functon at a grd pont x, s approxmated by a lnear sum of all functonal values on the whole doman and the quadrature formula for frst dervatve s gven as follows: df fx( x ) wj f ( x j ), 1,,, (4) dx j 1 where j x w represents the weghtng coeffcents to be evaluated, s the number of grd ponts [18]. The n-th order dervatve s defned as same dea gven by, ( n) j df f ( x ) w f ( x ), 1,,, ( n) ( n) x j j dx x j 1 where w represents the weghtng coeffcents, s the number of grd ponts. The man dea accordng to the procedure s to determne weghtng coeffcents. The DQ method offers usng unform or nonunform selecton of grd ponts but, t gves more effectve and stable solutons usng Chebyshev-Gauss Labotto ponts [18, 19]. Here, we choose grd ponts as the Chebyshev collocaton ponts defned as x cos( ),, 1,,, whch s applcable for only nterval [1,-1]. If the problem s gven on nterval [ ab, ] to obtan followng transformaton s used [18]. b a x (1 ) a. (7) When Lagrange nterpolatng polynomals are consdered as test functon, (5) (6) x where rk () x represents the test functon and K( x) rk ( x), 1,,, ( x x ) K ( x ) k k (8) K( x) ( x x )( x x )( x x ) ( x x ), K ( x) ( x x ). (9) 1 1 j k k 1, k j Usng polynomal approach theory, for frst order dervatve the weghtng coeffcents, gven n Eq. (4) becomes as follows [18]: K ( x ) aj ( x), j. ( x x ) K ( x ) j j For the entres on man dagonal, the followng relaton becomes, K ( x ) a ( x). ( x x ) K ( x ) j (10) (11)

4 By usng the lnear vector space spannng property as beng represented by dfferent knd of bases, on dagonal entres followng formula can be used whch s obtaned by power functons, k x 1, k 1,,, when k 1, a 0, a a. (1) j j j 1 j 1, j ow, to obtan quadrature solutons of the model problem Chebyshev polynomal s used as a bass together wth Chebyshev collocaton ponts, the functon Kx ( ) can be obtaned as, where where (1) K( x) (1 x ) T ( x ) (13) T (1) ( x) represents frst dervatve of T ( x) cos( ), and arccos( x ). Thus, The expresson can be rewrtten, T (1) ( x) (cos( )) sn( ) d (14) d x sn( ) (15) K( x) K( ) sn( )sn( ). (16) Snce, dervatve approxmaton s needed accordng to the structure of the method, by dfferentatng Eq. (16), or, K (1) ( x) ( sn( )sn( )) (17) (1) sn( )cos( ) sn( )cos( ) K ( x ). (18) sn( ) Snce, when sn( ) 0that s 0,, then Eq. (18) becomes (1) 1 K ( x ) ( 1). (19) When sn( ) 0, to remove the undetermned form L'Hosptal's rule s appled: (1) 0 K ( x ), (1) 1 K ( x ) ( 1). The reduced formulaton related to fnd frst dervatve matrx nterpreted as follows [18], [3]: (0) c ( 1) wj, 0, j, j, c ( x x ) x w, 1 1, (1 x ) w 00 j j w j 1 6 (1)

5 where c0 c and c j 1, 0 1. The model equaton requres rewrtng hgher order dervatves n DQ formulatons. In ths manner, matrx multplcaton method s used whch mentoned n [18]. 3. Quadrature Dscretzaton of the Model Equaton The KS equaton s rewrtten as Ut UU x U xx U xxxx () To obtan the approxmated soluton, we apply the method for each grd ponts, as follows, Ut ( x ) U ( x ) U x( x ) U xx( x ) U xxxx ( x ) (3) Then, spatal dervatves are replaced by the Dfferental Quadrature equalty, U( x ) t (1) () (4) j 1 j j j 1 j j j 1 j j U ( x ) w U ( x ) w U ( x ) w U ( x ) Frst order temporal dscretzaton s obtaned by forward Euler scheme, (4) U ( x, t) t U n 1 U t n (5) Matrx stablty has been studed for the DQ dscretzed systems. For a dscrete tme-dependent problem s of the form, U ( U ) t (6) wth proper ntal and boundary condtons. Here, represents spatal nonlnear dfferental operator. After applyng DQM and lnearzaton of the nonlnear term U( x) U ( x ) the equaton becomes, du { } dt [ A]{ U} { g} where { U } s an unknown vector of the functon values n the doman, { g } s the vector contanng the nonhomogeneous part and the boundary condtons and A s the dscretzed coeffcent matrx. The stablty of the numercal dscretzed system depends on egenvalues dstrbuton [18]. The condton for absolute stablty of the forward scheme s gven by, x (7) 1 t 1 (8) The stablty regon for the scheme s the crcle wth radus 1 and center (-1,0) on the complex t plane [30]. When the the dea s mplemented for the test problem 1, the maxmum real parts of the egenvalues are determned as , , , and , for 10, 0, 30, and 60 respectvely. Egenvalue dstrbutons for each grd ponts are gven by Fg The approxmated solutons are obtaned by combnng the explct scheme and and quadrature scheme by reducng the model to an algebrac system of equatons. The soluton of the matrx equaton gves desred soluton. Here, we used two sample problems to llustrate the effcency of the presented method and all the results n terms of error norms are gven n tables. In the end, solutons are also compared wth prevous studes. Solutons show approxmately same accurateness.

6 4. umercal Illustratons Effcency of the method s demonstrated usng L error norm whch s gven by, ex nu ex j nu j j 1 1/ L U U ( U ) ( U ) 1,,, (9) and maxmum error norm gven by L U U max ( U ) ( U ) 1,,,. (30) ex nu ex j nu j j To compare the accuracy wth prevous studes t s also measured global relatve error whch s gven by where ( Uex ) j ( Unu ) j j 1 GRE 1,,, ( U ) j 1 U ex and U nu represents the analytcal and numercal solutons, respectvely. ex Test Problem 1: As a frst case, KS equaton s consdered for 1 and 1. Exact soluton of the problem s [5]. j u( x, t) b 9tanh( k( x bt x0)) 11tanh ( k( x bt x 0)). (3) The ntal condton can be computed by takng exact soluton together wth boundary condtons gven by (1). umber of parttons are consdered 15, 30, 60 and 150. Comparsons between the exact and numercal solutons are tabulated for the nterval [0,1]. The soluton models the shock wave propagaton wth speed b and ntal poston x 0. Solutons are gven by Tab. 1. (31) Fgure 1- Egenvalue Dstrubuton when =10 Fgure - Egenvalue Dstrubuton when =0

7 Fgure 3- Egenvalue Dstrubuton when =30 Fgure 4- Egenvalue Dstrubuton when =60 Table 1- Error orms for Problem 1 Error orms t t 0.01 t E E E-03 L E E E E E E E E E E E E-03 L E E E E E E E E E-03 Fgure 4- Comparson between numercal and exact solutons of KS equaton when 150 Problem 1 (dotted lne represents numercal solutons)

8 Test Problem : ow, we consder the problem for 1 and 1. Exact soluton of the problem s gven by [5] u( x, t) b 3tanh( k( x bt x0)) 11tanh ( k( x bt x 0)) (33) The ntal condton s obtaned by takng the exact soluton together wth boundary condtons gven by. We have studed the algorthm wth parameters b 5, k 1/ 19, x 0 5. umber of parttons are consdered 15 and 30 and 00. Comparsons between the exact and numercal solutons are tabulated for the nterval [ 50,50]. Table - Error orms for Problem Error orms t t 0.01 t E E E-05 L E E E E E E E E E-06 L E E E E E E-06 Fgure 5- Comparson between numercal and exact solutons of KS equaton when 00 for Problem (dotted lne represents numercal solutons) Table 3- Comparson of Global Relatve Errors for Problem 1 Method GRE Chebyshev Dfferental Quadrature 1.194E-03 Exponental B-Splne Collocaton (Ersoy and Dag 016) [6] E-04 Quntc B-Splne Collocaton (Mttal and Arora 010) [5] 3.817E-04 Lattce Boltzmann (La and Ma 009) [31] 6.793E-04

9 Table 4- Comparson of Global Relatve Errors for Problem Method GRE Chebyshev Dfferental Quadrature 1.681E-05 Exponental B-Splne Collocaton (Ersoy and Dag 016) [6] E-05 Quntc B-Splne Collocaton (Mttal and Arora 010) [5] 6.509E-06 Lattce Boltzmann (La and Ma 009) [31] E Concluson In ths study, the Chebyshev based dfferental quadrature method s used for solutons of Kuramoto- Svashnksy equaton. The effcency of the approach s examned by two examples. Accordng to the Tab. 1 and Tab. as t ncreases, accuracy decreases, but as the number of parttons ncreases, we can see approxmately same accuracy. Also by Tab. 3 and Tab. 4, comparson wth other methods shows that effectveness s approxmately same for smlar numercal technques. It can be also seen that from the fgures numercal and exact solutons are n good agreement. The method s easy to mplement by usng small number of grd ponts. References [1] Kuramoto, Y., Tsuzuk, T., Persstent propagaton of concentraton waves n dsspatve meda far from thermal equlbrum, Progress of theoretcal physcs, 55 (1976),, pp [] Kuramoto, Y., and Tsuzuk T., On the Formaton of Dsspatve structures n reacton-dffuson systems, Progress of theoretcal physcs, 54 (1975), 3, pp [3] Svashnsky, G.I., Instabltes, pattern-formaton, and turbulence n flames, Annual Revew of Flud Mechancs, 15 (1983), 1, pp [4] Khater, A.H., Temsah, R.S., umercal solutons of the generalzed Kuramoto-Svashnsky equaton by Chebyshev spectral collocaton methods, Computers and Mathematcs wth Applcatons, 56 (008), 6, pp [5] Mttal, R.C., Arora, G., Quntc B-splne collocaton method for numercal soluton of the Kuramoto-Svashnsky equaton, Communcatons n onlnear Scence and umercal Smulaton, 15 (010), 10, pp [6] Ersoy, O., Dag, I., The exponental cubc B-Splne collocaton method for the Kuramoto- Svashnsky equaton, Flomat, 30 (016), 3, pp [7] Akrvs, G.D., Fnte dfference dscretzaton of Kuramoto-Svashnsky equaton, umersche Mathematk, 63 (199), 1, pp.1-11 [8] Xu, Y., Shu C.W., Local dscontnuous Galerkn methods for the Kuramoto-Svashnsky equatons and the Ito-type coupled KdV equatons, Computer methods n appled mechancs and engneerng, 195 (006), 5-8, pp [9] Uddn, M., Haq, S., Islam, S., A mesh-free numercal method for soluton of the famly of Kuramoto-Svashnsky equatons, Appled Mathematcs and Computaton, 1 (009),, pp

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