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Journal of Scientific Computing

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04-07-2018

Finite Element Methods for a System of Dispersive Equations

Authors: Jerry L. Bona, Hongqiu Chen, Ohannes Karakashian, Michael M. Wise

Published in: Journal of Scientific Computing | Issue 3/2018

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Abstract

The present study is concerned with the numerical approximation of periodic solutions of systems of Korteweg–de Vries type, coupled through their nonlinear terms. We construct, analyze and numerically validate two types of schemes that differ in their treatment of the third derivatives appearing in the system. One approach preserves a certain important invariant of the system, up to round-off error, while the other, somewhat more standard method introduces a measure of dissipation. For both methods, we prove convergence of a semi-discrete approximation and highlight differences in the basic assumptions required for each. Numerical experiments are also conducted with the aim of ascertaining the accuracy of the two schemes when integrations are made over long time intervals.

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Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. The Applied Mathematics Series, vol. 55. National Bureau of Standards, Gaithersburg (1965)

Adams, R.: Sobolev Spaces. Academic Press, New York (1975)MATH

Ash, J.M., Cohen, J., Wang, G.: On strongly interacting internal solitary waves. J. Fourier Anal. Appl. 2, 507–517 (1996)MathSciNetCrossRef

Baker, G.A., Dougalis, V.A., Karakashian, O.A.: Convergence of Galerkin approximations for the Korteweg–de Vries equation. Math. Comput. 40, 419–433 (1983)MathSciNetCrossRef

Bona, J.L.: Convergence of periodic wave trains in the limit of large wavelength. Appl. Sci. Res. 37, 21–30 (1981)MathSciNetCrossRef

Bona, J.L., Chen, H., Karakashian, O.A.: Stability of solitary-wave solutions of systems of dispersive equations. Appl. Math. Optim. 75, 27–53 (2017)MathSciNetCrossRef

Bona, J.L., Chen, H., Karakashian, O.A.: Instability of solitary-wave solutions of systems of coupled KdV equations: theory and numerical results, Preprint

Bona, J.L., Chen, H., Karakashian, O.A., Xing, Y.: Conservative, discontinuous-Galerkin methods for the generalized Korteweg–de Vries equation. Math. Comput. 82, 1401–1432 (2013)MathSciNetCrossRef

Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: part I. Derivation and linear theory. J. Nonlinear Sci. 12, 283–318 (2002)MathSciNetCrossRef

10.

Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: Part II. Nonlinear theory. Nonlinearity 17, 925–952 (2004)MathSciNetCrossRef

11.

Bona, J.L., Cohen, J., Wang, G.: Global well-posedness of a system of KdV-type with coupled quadratic nonlinearities. Nagoya Math. J. 215, 67–149 (2014)MathSciNetCrossRef

12.

Bona, J.L., Dougalis, V.A., Mitsotakis, D.: Numerical solutions of KdV–KdV systems of Boussinesq equations. I. The numerical scheme and generalized solitary waves. Math. Comput. Simul. 74, 214–226 (2007)MathSciNetCrossRef

13.

Bona, J.L., Dougalis, V.A., Mitsotakis, D.: Numerical solutions of KdV–KdV systems of Boussinesq equations. II. Evolution of radiating solitary waves. Nonlinearity 21, 2825–2848 (2008)MathSciNetCrossRef

14.

Bona, J.L., Smith, R.: The initial-value problem for the Korteweg–de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A 278, 555–601 (1975)MathSciNetCrossRef

15.

Bona, J.L., Soyeur, A.: On the stability of solitary-wave solutions of model equations for long waves. J. Nonlinear Sci. 4, 449–470 (1994)MathSciNetCrossRef

16.

Brenner, S., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2002)CrossRef

17.

Chen, H.: Long-period limit of nonlinear dispersive waves. Differ. Integral Equ. 19, 463–480 (2006)MathSciNetMATH

18.

Ciarlet, P.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1980)MATH

19.

Cheng, Y., Shu, C.-W.: A discontinuous finite element method for time dependent partial differential equations with higher order derivatives. Math. Comput. 77, 699–730 (2008)MathSciNetCrossRef

20.

Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Sharp global well-posedness of KdV and modified KdV on \({\mathbb{R}}\) and \({\mathbb{T}}\). J. Am. Math. Soc. 16, 705–749 (2003)MathSciNetCrossRef

21.

Dekker, K., Verwer, J.G.: Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations. North Holland, Amsterdam (1984)MATH

22.

Gear, J.A., Grimshaw, R.: Weak and strong interactions between internal solitary waves. Stud. Appl. Math. 65, 235–258 (1984)MathSciNetCrossRef

23.

Hakkaev, S.: Stability and instability of solitary wave solutions of a nonlinear dispersive system of Benjamin–Bona–Mahony type. Serdica Math. J. 29, 337–354 (2003)MathSciNetMATH

24.

Kappeler, T., Topalov, P.: Global well-posedness of KdV in \(H^{-1}({\mathbb{T}}, {\mathbb{R}})\). Duke Math. J. 135, 327–360 (2006)MathSciNetCrossRef

25.

Karakashian, O.A., Makridakis, C.: A posteriori error estimates for the generalized Korteweg–de Vries equation. Math. Comput. 84, 1145–1167 (2015)CrossRef

26.

Karakashian, O.A., Xing, Y.: A posteriori error estimates for the Generalized Korteweg–de Vries equation. Commun. Comput. Phys. 20, 250–278 (2016)MathSciNetCrossRef

27.

Liu, H., Yi, N.: A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg–de Vries equation. J. Comput. Phys. 321, 776–796 (2016)MathSciNetCrossRef

28.

Majda, A., Biello, J.: The nonlinear interaction of barotropic and equatorial baroclinic Rossby Waves. J. Atmos. Sci. 60, 1809–1821 (2003)MathSciNetCrossRef

29.

Miura, R.M.: The Korteweg–de Vries equation: a survey of results. SIAM Rev. 18, 412–459 (1976)MathSciNetCrossRef

30.

Pego, R.L., Weinstein, M.I.: Asymptotic stability of solitary waves. Commun. Math. Phys. 164, 305–349 (1994)MathSciNetCrossRef

31.

Wahlbin, L.B.: Dissipative Galerkin method for the numerical solution of first order hyperbolic equations. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, Proceedings of a Symposium at the Mathematics Research Center, the University of Wisconsin Madison, pp. 147-169. Academic Press (1974)

32.

Winther, R.: A conservative finite element method for the Korteweg–de Vries equation. Math. Comput. 34, 23–43 (1980)MathSciNetMATH

33.

Xu, Y., Shu, C.-W.: Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection–diffusion and KdV equations. Comput. Methods Appl. Mech. Eng. 196, 3805–3822 (2007)MathSciNetCrossRef

34.

Yan, J., Shu, C.-W.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40, 769–791 (2002)MathSciNetCrossRef

Title: Finite Element Methods for a System of Dispersive Equations
Authors: Jerry L. Bona
Hongqiu Chen
Ohannes Karakashian
Michael M. Wise
Publication date: 04-07-2018
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0767-x

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