Error estimate for a fully discrete spectral scheme for Korteweg-de Vries-Kawahara equation

• Autorzy: Ujjwal Koley
• Języki publikacji: EN

Abstrakty

EN

We are concerned with convergence of spectral method for the numerical solution of the initial-boundary value problem associated to the Korteweg-de Vries-Kawahara equation (Kawahara equation, in short), which is a transport equation perturbed by dispersive terms of the 3rd and 5th order. This equation appears in several fluid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in fluid dynamics. These equations are discretized in space by the standard Fourier-Galerkin spectral method and in time by the explicit leap-frog scheme. For the resulting fully discrete, conditionally stable scheme we prove an L 2-error bound of spectral accuracy in space and of second-order accuracy in time.

Słowa kluczowe

EN

Kawahara equation; Fourier-Galerkin spectral method; Error estimate; Convergence

Kategorie tematyczne

• 65T99: None of the above, but in this section
• 65G99: None of the above, but in this section
• 65M99: None of the above, but in this section

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 1
• Strony: 173-187

Daty

wydano

2012-02-01

online

2011-12-09

Twórcy

autor

Ujjwal Koley

• University of Oslo

Bibliografia

• [1] Baker G.A., Dougalis V.A., Karakashian O.A., Convergence of Galerkin approximations for the Korteweg-de Vries equation, Math. Comp., 1983, 40(162), 419–433 http://dx.doi.org/10.1090/S0025-5718-1983-0689464-4
• [2] Bona J.L., Dougalis V.A., Karakashian O.A., Fully discrete Galerkin methods for the Korteweg-de Vries equation, Comput. Math. Appl. Ser. A, 1986, 12(7), 859–884 http://dx.doi.org/10.1016/0898-1221(86)90031-3
• [3] Ceballos J.C., Sepúlveda M., Vera Villagrán O.P., The Korteweg-de Vries-Kawahara equation in a bounded domain and some numerical results, Appl. Math. Comput., 2007, 190(1) 912–936 http://dx.doi.org/10.1016/j.amc.2007.01.107
• [4] Cui S.B., Deng D.G., Tao S.P., Global existence of solutions for the Cauchy problem of the Kawahara equation with L 2 initial data, Acta Math. Sin. (Engl. Ser.), 2006, 22(5), 1457–1466 http://dx.doi.org/10.1007/s10114-005-0710-6
• [5] Darvishi M.T., New algorithms for solving ODEs by pseudospectral method, Korean J. Comput. Appl. Math., 2000, 7(2), 319–331
• [6] Darvishi M.T., Preconditioning and domain decomposition schemes to solve PDEs, Int. J. Pure Appl. Math., 2004, 15(4), 419–439
• [7] Darvishi M.T., Khani F., Numerical and explicit solutions of the fifth-order Korteweg-de Vries equations, Chaos Solitons Fractals, 2009, 39(5), 2484–2490 http://dx.doi.org/10.1016/j.chaos.2007.07.034
• [8] Darvishi M.T., Khani F., Kheybari S., Spectral collocation solution of a generalized Hirota-Satsuma coupled KdV equation, Int. J. Comput. Math., 2007, 84(4), 541–551 http://dx.doi.org/10.1080/00207160701227863
• [9] Darvishi M.T., Kheybari S., Khani F., A numerical solution of the Korteweg-de Vries equation by pseudospectral method using Darvishi’s preconditionings, Appl. Math. Comput., 2006, 182(1), 98–105 http://dx.doi.org/10.1016/j.amc.2006.03.039
• [10] Darvishi M.T., Kheybari S., Khani F., Spectral collocation method and Darvishi’s preconditionings to solve the generalized Burgers-Huxley equation, Commun. Nonlinear Sci. Numer. Simul., 2008, 13(10), 2091–2103 http://dx.doi.org/10.1016/j.cnsns.2007.05.023
• [11] Doronin G.G., Larkin N.A., Well and ill-posed problems for the KdV and Kawahara equations, Bol. Soc. Parana. Mat., 2008, 26(1–2), 133–137
• [12] Gorsky J., Himonas A.A., Well-posedness of KdV with higher dispersion, Math. Comput. Simulation, 2009, 80(1), 173–183 http://dx.doi.org/10.1016/j.matcom.2009.06.007
• [13] Hunter J.K., Scheurle J., Existence of perturbed solitary wave solutions to a model equation for water waves, Phys. D, 1988, 32(2), 253–268 http://dx.doi.org/10.1016/0167-2789(88)90054-1
• [14] Kato T., On the Korteweg-de Vries equation, Manuscripta Math., 1979, 28(1–3), 89–99 http://dx.doi.org/10.1007/BF01647967
• [15] Kato T., On the Cauchy problem for the (generalized) Korteweg-de Vries equations, In: Studies in Applied Mathematics, Adv. Math. Suppl. Stud., 8, Academic Press, New York, 1983, 93–128
• [16] Kawahara T., Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 1972, 33(1), 260–264 http://dx.doi.org/10.1143/JPSJ.33.260
• [17] Kichenassamy S., Olver P.J., Existence and nonexistence of solitary wave solutions to higher-order model evolution equations, SIAM J. Math. Anal., 1992, 23(5), 1141–1166 http://dx.doi.org/10.1137/0523064
• [18] Koley U., Convergence of numerical schemes for the Korteweg-de Vries-Kawahara equation (submitted)
• [19] Maday Y., Quarteroni A., Error analysis for spectral approximation of the Korteweg-de Vries equation, RAIRO Modél. Math. Anal. Numér., 1988, 22(3), 499–529
• [20] Mercier B., An Introduction to the Numerical Analysis of Spectral Methods, Lecture Notes in Phys., 318, Springer, Berlin, 1989
• [21] Pelloni B., Dougalis V.A., Error estimates for a fully discrete spectral scheme for a class of nonlinear, nonlocal dispersive wave equations, Appl. Numer. Math., 2001, 37(1–2), 95–107 http://dx.doi.org/10.1016/S0168-9274(00)00027-1
• [22] Ponce G., Regularity of solutions to nonlinear dispersive equations, J. Differential Equations, 1989, 78(1), 122–135 http://dx.doi.org/10.1016/0022-0396(89)90078-8
• [23] Sepúlveda M., Vera Villagrán O.P., Numerical method for a transport equation perturbed by dispersive terms of 3rd and 5th order, Sci. Ser. A Math. Sci. (N.S.), 2006, 13, 13–21
• [24] Temam R., Navier-Stokes Equations. Theory and Numerical Analysis, revised ed., Stud. Math. Appl., 2, North-Holland, Amsterdam-New York-Oxford, 1979
• [25] Vera Villagrán O.P., Gain of regularity for a Korteweg-de Vries-Kawahara type equation, Electron. J. Differential Equations, 2004, #71
• [26] Wang H., Cui S.B., Deng D.G., Global existence of solutions for the Kawahara equation in Sobolev spaces of negative indices, Acta Math. Sin. (Engl. Ser.), 2007, 23(8), 1435–1446 http://dx.doi.org/10.1007/s10114-007-0959-z

Identyfikatory

DOI

10.2478/s11533-011-0055-6