COMPACT SCHEMES FOR KORTEWEG-DE VRIES EQUATION

Transcription

1 THERMAL SCIENCE, Year 07, Vol., No. 4, pp COMPACT SCHEMES FOR KORTEWEG-DE VRIES EQUATION by Xiu-Lig YIN a,b*, Cheg-Jia ZHANG a*, Jig-Jig ZHANG c, ad Ya-Qi LIU b a School of Mathematics ad Statistics, Huazhog Uiversity of Sciece ad Techology, Wuha, Chia b School of Mathematical Scieces, Dezhou Uiversity, Dezhou, Chia c School of Sciece, East Chia Jiaotog Uiversity, Nachag, Jiagxi, Chia Origial scietific paper This paper proposes oe family of compact schemes for Korteweg-de Vries equatio. I the determiistic case, the schemes are coverget with fourth-order accuracy both i space ad i time. Moreover, the schemes are stable. The umerical dispersio relatio is aalyzed. We compare the schemes with oe secod-order scheme. The umerical examples test the effect of the schemes. I the stochastic case, we simulate the wave profile ad three discrete dyamical quatities for Korteweg-de Vries equatio with small oise. The white oise has stochastic ifluece o the profile ad dyamical quatities of the solutio. If the size of oise icreases, the perturbatio o the profile ad dyamical quatities will icrease accordigly. Key words: Korteweg-de Vries equatio, compact scheme, stability Itroductio Korteweg-de Vries (KdV) equatio ad its modificatios are widely used to study shallow water wave with small amplitude or with weakly o-liear restorig force, MHD wave i collisio less plasma, bubble-liquid mixig wave, ad o-liear log wave i oharmoic lattice []. The KdV equatios describe the iteractio ad balace of dispersio term ad o-liear term. There exist stable solitos ad recostructio of iitial waveform for KdV equatios. Some theoretical aalysis ad umerical simulatio for KdV equatios have bee proposed, for examples, the variatioal iteratio method [-4], the homotopy perturbatio method [, 5], the exp-fuctio method [6, 7], ad first itegral method [8], symplectic scheme [9, 0], fiite elemet method [], the meshless method of lies [], ad fiite differece method []. Recetly compact schemes, which possess high accuracy, compactess, ad ecoomic resource, are widely used i scietific computatio [4-8]. A compact splittig multisymplectic scheme for some Schrodiger equatios is proposed i [5]. Kaazawa et al. [7] ivestigates a coservative compact fiite differece scheme for the determiistic KdV equatio. A pseudo-compact scheme for the Roseau-KdV equatio couplig with the Roseau- RLW equatio is aalyzed i [8]. * Correspodig authors, s: yillmm@6.com; cjzhag@mail.hust.edu.c

2 798 THERMAL SCIENCE, Year 07, Vol., No. 4, pp It is foud that, for stochastic KdV equatios, the referece of compact schemes is lackig. The compact scheme is valid for determiistic KdV equatio ad its theory has already bee well established [7, 8], but what will happe for its stochastic parter? How will the compact scheme behave? The two questios motivate us to desig efficiet schemes for stochastic KdV equatio. By applyig compact operators, we costruct oe-parameter family of compact schemes to a iitial value problem of the followig stochastic KdV equatio: ut + uux + ε uxxx µχ, x [ 0, L] () ux (,0) gx ( ), t [ 0, T] where g(x) is a differetial fuctio, ut, u x, ad u xxx are the mea partial derivatives of u with respect to t ad x, respectively, μ is a small real umber, ad χ a real-valued white oise which is delta correlated i time, either smooth or delta correlated i space. For KdV equatios, uu x, ε u xxx, ad µχ are o-liear term, dispersio term, ad stochastic disturbace term, respectively. If µ 0, the system () is a determiistic system ad preserves some physical quatities [9] uder periodic coditio: L L L ε x () H( t) uxt (, ) u( xt, ) d x, H ( t) uxt (, ) d x, H( t) uxt (, )dx These physical quatities ca be used to test the efficiecy of umerical methods for the stochastic KdV equatio. Compact schemes Further i the text, by applyig a kid of compact operators, we preset a kid of compact schemes for KdV equatio (). For simplicity, we cosider the uiform mesh grids { xk kh, t τ } with step-sizes h L/N ad τ T/M. Numerical values of ux ( k, t ) are deoted by u, u k, ad u stad for solutio vectors at x x k ad t t with compoets k ux ( k,) t ad uxt (, ), respectively. For spatial discretizatio of eq. () uder periodic coditio, we cosider the liear operators Auk αuk + uk + αuk+ ad: u k + u k u k + u k u k, k k + u k + + u k u Bu b a Cu a k + () 4h h h We adopt the operators: xuk A Buk, δxuk A Cuk δ (4) to approximate u x ad u xxx, respectively [4, 5]. Deote four cyclic matrices by: 0 k k k k α α k 0 k k k α α k k 0 k k M, M0( k, k) (5) α α k k 0 k k α α k k k 0 k N N k k k k 0 N N

3 THERMAL SCIENCE, Year 07, Vol., No. 4, pp M M ( a/ h, b/4 h), M ( a/ h ) M (,). The, uder periodic coditios, the matrix form of (4) is: δ xuk M Muk, δxuk M Muk (6) Oe-parameter family of fourth-order approximatios δ x to u x are defied by a α, b 4α. Whe a, α 0.5, a fourth order approximatio δ x to u xxx is predicted. Taylor aalysis yields that, the leadig trucatio errors of δ x ad δ x are 4/5!(α )(u xxxxx) kh 4 ad 4/7!(u xxxxxx) kh 4, respectively. The Fourier aalysis yields that: where δ u iv u, δ u iv u (7) x k k x k k si( ) si( ) si( )[cos( ) ] v a βh + b βh, v a βh βh h[ + αcos( βh)] h [ + αcos( βh)] Discretizig KdV equatios () i spatial directio with compact operators (4) yields the followig ordiary differetial equatio of zt ( ) ux ( k, t): z + zδ z+ εδ z µχ (9) x x k By applyig famous 4-stage Ruge-Kutta method [0] to eq. (9) we ca get: z + τ z + [ f( T, Y) + f( T, Y)] Y z + τ f( T, Y) + f( T, Y) Y z + τ + f( T, Y) + f( T, Y) (0) where ftz (, ) zδ xz εδ xz+ µχ k, T t + ( ) τ /6, T t + ( + ) τ /6. Determiistic aalysis Cosider the determiistic case that µ 0. Suppose that u k ux ( k, t). Taylor aalysis yields that the compact schemes (0) are coverget with 4-order accuracy both i time ad i space. The schemes are most compact tridiagoal schemes with miimal spatial grid. The leadig terms of local trucatio errors of (0) are: !(α )( uxxxxx ) k h +!( uxxxxxxx ) k h ( 4) ( ) 4 4 f f f f f f f f f f ff + τ + + (5 ) k Cosider the compact schemes (0) applied to the liear case ut + pux + quxxx 0. By computig we get that: τ pδx qδx τ pδx qδx τ pδx qδx τ pδx qδx + ( + ) 6 ( + ) + u u ( + ) + 6 ( + ) + (8)

4 800 THERMAL SCIENCE, Year 07, Vol., No. 4, pp Sice the eigevalues of compact operators δ x, δ x are zeros or imagiary umbers, the modules of multiplyig matrices are ad so the schemes (0) are liearly stable. The we cosider the dispersio relatio i liear case. Suppose that the exact solutio is i the form: ξ ω uxt (, ) exp i x+ t h τ where ξ is the wave umber ad ω the frequecy. The by isertig it ito the liear equatio we ca get the exact dispersio relatio: c ω + cξ ξ 0 () d where c pτ/h ad d ph /q. Assume that the formal umerical solutio is u k exp[ ik ( ξ + ω)]. By computig accordig to eq. (0), we get the followig umerical dispersio relatio: AA ( ) siω () 4 A + A + 44 where c (si ξ si ξ) siξ A + c d + cosξ + cosξ Secod-order scheme If b 0, a, ad α 0, the the compact operators δx ad δxare secod-order with the leadig trucatio errors (/6)( uxxx ) k h ad (/4)( uxxxxx ) k h, respectively. By applyig the forward differece formula of first derivative ad the averagig operator to eq. (9), we ca get that: + uk uk + / + / + / + ( uδ xu) k + εδ xuk µχ k () τ where / / / u + ( ), k u + k + uk χ + k χ + k hτ / where χ + k is a sequece of idepedet radom variables with ormal distributio law N(0, ) []. This meas that we discretize eq. (9) i temporal directio: + + / + / + / k k τ ( δ x ) k τεδx k τµχk u k uxk t u u + u u + u (4) Cosider the determiistic case that μ 0. Suppose that (, ). Taylor aalysis yields that the scheme () is coverget with secod-order accuracy both i time ad i space. The leadig terms of local trucatio errors of eq. () is: xxx k xxxxx k k τ ( u ) h + ( u ) h + ( ff f f) 6 4 4

5 THERMAL SCIENCE, Year 07, Vol., No. 4, pp which meas that: The scheme (4) applied to liear equatio geerates the formula: + τδ + x τδ + x ( u u ) + p ( u + u ) + q ( u + u ) 0 c ( + ) ( + + ) k k k k k k k 4u + c u u + u u + u u d c ( + ) ( + + ) k k k k k k k 4u cu u u u + u u d Therefore, the module of multiplyig matrix is ad so the scheme (4) is liearly stable. The we cosider the dispersio relatio i liear case. By computig accordig to eq. (4), we get umerical dispersio relatio: 4B siω (5) B + 4 where B csi ξ + ( cd / )(si ξ si ξ). No-periodic case Now we cosider spatial discretizatio uder o-periodic coditio. At some discrete poits ear to the boudary, we apply the followig compact operators for scheme (0): δ 7u 9u 9u u4 7uN 9uN 9uN un xu δ + + xu, xun xun, 6h δ δ h δxu 5 δxu (u 76u + 98u 58u4 + 6u5 u6), h δxun 5 δxun ( u 76 N + un 98uN + 58uN 6uN 4 + u N 5), h δxu + δxu δxu ( 0 u + 5 u 46 u + 8 u4 8 u5 + u6 ), 4 4 7h δxun + δxun δxun (0 un 5 un + 46 un 8 un + 8 un 4 u N 5 ) 4 4 7h Meawhile, for scheme (4), we give the discrete spatial operator at boudary poits: u + 4u u un 4uN + un δxu, δxun, h h δxu ( 5 u + 8 u 4 u + 4 u4 u5 ), h δxun (5 un 8 un + 4 un 4 un + un 4 ), h δxu ( u + 0 u u + 6 u4 u5 ), h δxun ( un 0 un + un 6 un + un 4 ) h

6 80 THERMAL SCIENCE, Year 07, Vol., No. 4, pp Numerical examples Now we apply the schemes (0) ad (4) to solve the KdV equatio () with: K ε 4.84e 4, L, K 0., K ε, g Ksec h( Kx 6) For the sake of simplicity, we choose α /4 for δ x to eq. (). Numerical results are similar to other differet α. First to test the efficiecy of the schemes i the determiistic case, we suppose that μ 0. I fact, i this determiistic case, the exact solutio of the system () is: u K sec h ( K x KKt 6) I fig. we depict the exact dispersio relatio () ad umerical dispersio relatios () ad (5) correspodig to c 0. ad differet d. The umerical dispersio curve of scheme (0), relative to scheme (4), is closer to the exact curve. Figure. Dispersio relatio curves d.98 (a), d 6. (b) I fig. we plot the secod-orm errors betwee umerical solutios ad exact solutio at t with h 0.05, τ The errors coform that the schemes (0) are more accurate tha the scheme (4). We defie three discrete dyamical quatities for H (t), H (t), ad H (t): (6) [( k) εδ ( x k) ], ( k), k k k k H h u u H h u H h u Deote the residual errors of H, H ad H by: H H H ( t ), H H H ( t ), H H H ( t ) (7) respectively. From figs. -5, we depict the residual errors H, H, ad H for umerical solutio with h 0.05, τ These figures show that for determiistic KdV equatio, the compact schemes (0) simulate the discrete dyamical quatities (6) more approximately tha the scheme (4).

7 THERMAL SCIENCE, Year 07, Vol., No. 4, pp Figure. Errors of umerical solutios for scheme (0) (b) ad scheme (4) (a) Figure. Errors of H for scheme (0) (b) ad scheme (4) (a) Figure 4. Errors of H for scheme (0) (b) ad scheme (4) (a) Next we simulate the profile of stochastic KdV wave with small μ. I fig. 6 we plot the -D profile of uk with scheme (0) for oe trajectory with h 0.05, τ 0.000, μ 0.05 (b), ad μ (a). From u k depicted i the figure, we fid that the white oise produces stochastic ifluece o the KdV waves. At last we simulate the stochastic ifluece of white oise o the physical quatities. From figs. 7-9, we depict the discrete quatities (6) for umerical solutio with (0) of oe trajectory with h 0.05, τ 0.000, μ 0.05 (b), ad μ (a).

8 804 THERMAL SCIENCE, Year 07, Vol., No. 4, pp Figure 5. Errors of H for scheme (0) (b) ad scheme (4) (a) Figure 6. The -D profile for μ 0.05 (b) ad μ (a) Figure 7. The H for umerical solutio with μ 0.05 (b) ad μ (a)

9 THERMAL SCIENCE, Year 07, Vol., No. 4, pp Figure 8. The H for umerical solutio with μ 0.05 (b) ad μ (a) Figure 9. The H for umerical solutio with μ 0.05 (b) ad μ (a) From the results depicted i figs. 7-9, we ca fid that the white oise produces stochastic perturbatio o the three discrete dyamical quatities. If the size of oise icreases, the perturbatio o the profile ad dyamical quatities will icrease accordigly. The larger the size of oise is, the more the perturbatio will be. Coclusio For KdV equatio, we preset compact schemes (0) with miimal spatial grids. They are stable ad coverget with fourth-order accuracy i each directio for determiistic KdV equatio. Compared with scheme (4), the schemes (0) are more efficiet (see figs. -5) i simulatig the solutios, dispersio relatio ad physical quatities. For stochastic equatio, the white oise has stochastic perturbatio upo the wave profile ad discrete dyamical quatities (see figs. 6-9). We observe that the perturbatio will icrease with the size of oise. Next, further discussio ad research about the strog ad weak covergece for the schemes will be eeded.

10 806 THERMAL SCIENCE, Year 07, Vol., No. 4, pp Ackowledgmet This work is supported by Natioal Natural Sciece Foudatio of Chia (Grat No. 5008, 05) ad Natural Sciece Foudatio of Shadog Provice (No. ZR05AL06, ZR06AQ07, J7KA56). Refereces [] Milstei, G., Tretyakov, M., Stochastic Numerics for Mathematical Physics, Kluwer Academic Publishers, Dordrecht, the Netherlads, 995 [] Cui, Q. N., et al., Aalytical ad Numerical Methods for Thermal Sciece, Thermal Sciece, 0 (06),, pp. IX-XIV [] Lu, J. F., Ma, L., Aalytical Approach to a Geeralized Hirota-Satsuma Coupled Korteweg-de Vries Equatio by Modified Variatioal Iteratio Method, Thermal Sciece, 0 (06),, pp [4] He, J.-H., A New Fractal Derivatio, Thermal Sciece, 5 (0), Suppl., pp. S45-S47 [5] Wag, Q., Homotopy Perturbatio Method for Fractioal KdV-Burgers Equatio, Chaos Solitos & Fractals, 5 (008), 5, pp [6] Zhag, S., et al., Multi-Wave Solutios for a No-Isospectral KdV-Type Equatio with Variable Coefficiets, Thermal Sciece, 6 (0), 5, pp [7] He, J.-H., Abdou, M. A., New Periodic Solutios for Noliear Evolutio Equatios Usig Exp- Fuctio Method, Chaos, Solitos & Fractals, 4 (007), 5, pp [8] Solima, A., Rasla, R., First Itegral Method for the Improved Modified KdV Equatio, Iteratioal Joural of Noliear Sciece, 8 (009),, pp. -8 [9] Lv, Z. Q., et al., A New Multi-Symplectic Scheme for the KdV Equatio, Chiese Physics Letters, 8 (0), 6, pp [0] Zhao, P. F., Qi, M. Z., Multisymplectic Geometry ad Multisymplectic Preissma Scheme for the KdV Equatio, Joural of Physics A: Mathematical ad Geeral, (000), 8, pp [] Caivar, A., et al., A Taylor-Galerki Fiite Elemet Method for the KdV Equatio Usig Cubic B- Splies, Physica B Codesed Matter, 405 (00), 6, pp [] She, Q., A Meshless Method of Lies for the Numerical Solutio of KdV Equatio Usig Radial Basis Fuctios, Egieerig Aalysis with Boudary Elemets, (009), 0, pp [] Hu, J., et al., Coservative Liear Differece Scheme for Roseau-KdV Equatio, Advaces i Mathematical Physics, 0 (0),, pp [4] Lele, S., Compact Fiite Differece Schemes with Spectral-Like Solutio, Joural of Computatioal Physics, 0 (99),, pp. 6-4 [5] Ma, Y. P., et al., High-Order Compact Splittig Multi-Symplectic Method for the Coupled Noliear Schroediger Equatios, Computer ad Mathematics with Applicatios, 6 (0),, pp. 9- [6] Sekhar, T., Raju, B., A Efficiet Higher Order Compact Scheme to Capture Heat Trasfer Solutios i Spherical Geometry, Computer Physics Commuicatios, 8 (0),, pp [7] Kaazawa, H., et al., A Coservative Compact Fiite Differece Scheme for the KdV Equatio, JSIAM Letters, 4 (0), Mar., pp. 5-8 [8] Pa, X. T., et al., Numerical Aalysis of a Pseudo-Compact C-N Coservative Scheme for the Roseau- KdV Equatio Couplig with the Roseau-RLW Equatio, Boudary Value Problems, (05), Dec., pp. -7 [9] Wag, Q., Variatioal Priciple for Variable Coefficiets KdV Equatio, Physics Letters A, 58 (006),, pp. 9-9 [0] Liu, X. S., et al., Dyamic Properties of the Cubic Noliear Schroediger Equatio by Symplectic Method, Chiese Physics, 5 (005),, pp. -7 [] Jiag, S. S., et al., Stochastic Multi-Symplectic Itegrator for Stochastic Hamiltoia Noliear Schroediger Equatio, Commuicatios i Computatioal Physics, 4 (0),, pp. 9-4 Paper submitted: Jue, 06 Paper revised: August 5, 06 Paper accepted: May 9, Society of Thermal Egieers of Serbia. Published by the Viča Istitute of Nuclear Scieces, Belgrade, Serbia. This is a ope access article distributed uder the CC BY-NC-ND 4.0 terms ad coditios.