A posteriori error estimates for discontinuous Galerkin methods for the generalized Korteweg-de Vries equation

Karakashian, Ohannes; Makridakis, Charalambos

doi:10.1090/S0025-5718-2014-02878-0

A posteriori error estimates for discontinuous Galerkin methods for the generalized Korteweg-de Vries equation
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Math. Comp. 84 (2015), 1145-1167 Request permission

Abstract:

We construct, analyze and numerically validate a posteriori error estimates for conservative discontinuous Galerkin (DG) schemes for the Generalized Korteweg-de Vries (GKdV) equation. We develop the concept of dispersive reconstruction, i.e., a piecewise polynomial function which satisfies the GKdV equation in the strong sense but with a computable forcing term enabling the use of a priori error estimation techniques to obtain computable upper bounds for the error. Both semidiscrete and fully discrete approximations are treated.

References

Kanji Abe and Osamu Inoue, Fourier expansion solution of the Korteweg-de Vries equation, J. Comput. Phys. 34 (1980), no. 2, 202–210. MR 559996, DOI 10.1016/0021-9991(80)90105-9
Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Georgios Akrivis, Charalambos Makridakis, and Ricardo H. Nochetto, A posteriori error estimates for the Crank-Nicolson method for parabolic equations, Math. Comp. 75 (2006), no. 254, 511–531. MR 2196979, DOI 10.1090/S0025-5718-05-01800-4
M. E. Alexander and J. Ll. Morris, Galerkin methods applied to some model equations for non-linear dispersive waves, J. Comput. Phys. 30 (1979), no. 3, 428–451. MR 530003, DOI 10.1016/0021-9991(79)90124-4
Garth A. Baker, Vassilios A. Dougalis, and Ohannes A. Karakashian, Convergence of Galerkin approximations for the Korteweg-de Vries equation, Math. Comp. 40 (1983), no. 162, 419–433. MR 689464, DOI 10.1090/S0025-5718-1983-0689464-4
Garth A. Baker, Wadi N. Jureidini, and Ohannes A. Karakashian, Piecewise solenoidal vector fields and the Stokes problem, SIAM J. Numer. Anal. 27 (1990), no. 6, 1466–1485. MR 1080332, DOI 10.1137/0727085
E. Bänsch, F. Karakatsani, and Ch. Makridakis, A posteriori error control for fully discrete Crank-Nicolson schemes, SIAM J. Numer. Anal. 50 (2012), no. 6, 2845–2872. MR 3022245, DOI 10.1137/110839424
T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A 272 (1972), no. 1220, 47–78. MR 427868, DOI 10.1098/rsta.1972.0032
Jerry Bona, Model equations for waves in nonlinear dispersive systems, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 887–894. MR 562704
J. Bona. On solitary waves and their role in the evolution of long waves, in H. Amann, N. Bazley, and K. Kirchgässner, editors, Applications of Nonlinear Analysis in the Physical Sciences, pages 183–205. Pitman: London, 1981.
J. L. Bona, H. Chen, O. Karakashian, and Y. Xing, Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation, Math. Comp. 82 (2013), no. 283, 1401–1432. MR 3042569, DOI 10.1090/S0025-5718-2013-02661-0
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376, DOI 10.1007/978-1-4757-3658-8
Yingda Cheng and Chi-Wang Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Math. Comp. 77 (2008), no. 262, 699–730. MR 2373176, DOI 10.1090/S0025-5718-07-02045-5
Bernardo Cockburn and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp. 52 (1989), no. 186, 411–435. MR 983311, DOI 10.1090/S0025-5718-1989-0983311-4
Vassilios A. Dougalis and Ohannes A. Karakashian, On some high-order accurate fully discrete Galerkin methods for the Korteweg-de Vries equation, Math. Comp. 45 (1985), no. 172, 329–345. MR 804927, DOI 10.1090/S0025-5718-1985-0804927-8
B. Fornberg and G. B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. Roy. Soc. London Ser. A 289 (1978), no. 1361, 373–404. MR 497916, DOI 10.1098/rsta.1978.0064
Emmanuil H. Georgoulis, Omar Lakkis, and Juha M. Virtanen, A posteriori error control for discontinuous Galerkin methods for parabolic problems, SIAM J. Numer. Anal. 49 (2011), no. 2, 427–458. MR 2784879, DOI 10.1137/080722461
Ohannes Karakashian and William McKinney, On optimal high-order in time approximations for the Korteweg-de Vries equation, Math. Comp. 55 (1990), no. 192, 473–496. MR 1035935, DOI 10.1090/S0025-5718-1990-1035935-4
Irene Kyza and Charalambos Makridakis, Analysis for time discrete approximations of blow-up solutions of semilinear parabolic equations, SIAM J. Numer. Anal. 49 (2011), no. 1, 405–426. MR 2783232, DOI 10.1137/100796819
Omar Lakkis and Charalambos Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems, Math. Comp. 75 (2006), no. 256, 1627–1658. MR 2240628, DOI 10.1090/S0025-5718-06-01858-8
Omar Lakkis and Tristan Pryer, Gradient recovery in adaptive finite-element methods for parabolic problems, IMA J. Numer. Anal. 32 (2012), no. 1, 246–278. MR 2875251, DOI 10.1093/imanum/drq019
Charalambos Makridakis and Ricardo H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J. Numer. Anal. 41 (2003), no. 4, 1585–1594. MR 2034895, DOI 10.1137/S0036142902406314
Charalambos Makridakis and Ricardo H. Nochetto, A posteriori error analysis for higher order dissipative methods for evolution problems, Numer. Math. 104 (2006), no. 4, 489–514. MR 2249675, DOI 10.1007/s00211-006-0013-6
J.-C. Saut, Applications de l’interpolation non linéaire à des problèmes d’évolution non linéaires, J. Math. Pures Appl. (9) 54 (1975), 27–52 (French). MR 454374
Hans Schamel and Klaus Elsässer, The application of the spectral method to nonlinear wave propagation, J. Comput. Phys. 22 (1976), no. 4, 501–516. MR 449164, DOI 10.1016/0021-9991(76)90046-2
Thiab R. Taha and Mark J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation, J. Comput. Phys. 55 (1984), no. 2, 231–253. MR 762364, DOI 10.1016/0021-9991(84)90004-4
T. Taha and M. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. IV, Numerical, modified Korteweg-de Vries equation. J. Comp. Phys., 77:540–548, 1988.
F. Tappert, Numerical solution of the Korteweg-de Vries equation and its generalizations by the split-step Fourier method. In A. C. Newell, editor, Nonlinear Wave Motion, Lectures in Applied Mathematics, pages 215–216. Amer. Math. Soc., Providence, R.I., 1974.
A. C. Vliegenthart, On finite-difference methods for the Korteweg-de Vries equation, J. Engrg. Math. 5 (1971), 137–155. MR 363153, DOI 10.1007/BF01535405
Lars B. Wahlbin, A dissipative Galerkin method for the numerical solution of first order hyperbolic equations, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 147–169. MR 0658322
Ragnar Winther, A conservative finite element method for the Korteweg-de Vries equation, Math. Comp. 34 (1980), no. 149, 23–43. MR 551289, DOI 10.1090/S0025-5718-1980-0551289-5
Yan Xu and Chi-Wang Shu, Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37-40, 3805–3822. MR 2340006, DOI 10.1016/j.cma.2006.10.043
Jue Yan and Chi-Wang Shu, A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal. 40 (2002), no. 2, 769–791. MR 1921677, DOI 10.1137/S0036142901390378