Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation
Authors:
J. L. Bona, H. Chen, O. Karakashian and Y. Xing
Journal:
Math. Comp. 82 (2013), 1401-1432
MSC (2010):
Primary 65N12, 65N30, 35Q35, 35Q51, 35Q53, 35Q86, 76B15, 76B25
DOI:
https://doi.org/10.1090/S0025-5718-2013-02661-0
Published electronically:
January 7, 2013
MathSciNet review:
3042569
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We construct, analyze and numerically validate a class of conservative, discontinuous Galerkin schemes for the Generalized Korteweg-de Vries equation. Up to round-off error, these schemes preserve discrete versions of the first two invariants (the integral of the solution, usually identified with the mass, and the -norm) of the continuous solution. Numerical evidence is provided indicating that these conservation properties impart the approximations with beneficial attributes, such as more faithful reproduction of the amplitude and phase of traveling-wave solutions. The numerical simulations also indicate that the discretization errors grow only linearly as a function of time.
- 1. L. Abdelouhab, J. L. Bona, M. Felland, and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D 40 (1989), no. 3, 360–392. MR 1044731, https://doi.org/10.1016/0167-2789(89)90050-X
- 2. 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, https://doi.org/10.1016/0021-9991(80)90105-9
- 3. Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
- 4. Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
- 5. A. A. Alazman, J. P. Albert, J. L. Bona, M. Chen, and J. Wu, Comparisons between the BBM equation and a Boussinesq system, Adv. Differential Equations 11 (2006), no. 2, 121–166. MR 2194497
- 6. J. P. Albert, J. L. Bona, and M. Felland, A criterion for the formation of singularities for the generalized Korteweg-de Vries equation, Mat. Apl. Comput. 7 (1988), no. 1, 3–11 (English, with Portuguese summary). MR 965674
- 7. 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, https://doi.org/10.1016/0021-9991(79)90124-4
- 8. J. Angulo, J. L. Bona, F. Linares, and M. Scialom, Scaling, stability and singularities for nonlinear, dispersive wave equations: the critical case, Nonlinearity 15 (2002), no. 3, 759–786. MR 1901105, https://doi.org/10.1088/0951-7715/15/3/315
- 9. Jaime Angulo Pava, Jerry L. Bona, and Marcia Scialom, Stability of cnoidal waves, Adv. Differential Equations 11 (2006), no. 12, 1321–1374. MR 2276856
- 10. 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, https://doi.org/10.1090/S0025-5718-1983-0689464-4
- 11. 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, https://doi.org/10.1137/0727085
- 12. T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London Ser. A 328 (1972), 153–183. MR 338584, https://doi.org/10.1098/rspa.1972.0074
- 13. 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, https://doi.org/10.1098/rsta.1972.0032
- 14. J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. London Ser. A 344 (1975), no. 1638, 363–374. MR 386438, https://doi.org/10.1098/rspa.1975.0106
- 15. 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
- 16. Jerry L. Bona, Convergence of periodic wavetrains in the limit of large wavelength, Appl. Sci. Res. 37 (1981), no. 1-2, 21–30. MR 633079, https://doi.org/10.1007/BF00382614
- 17.
J.L. 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. - 18. J. L. Bona, H. Chen, S.-M. Sun, and B.-Y. Zhang, Approximating initial-value problems with two-point boundary-value problems: BBM-equation, Bull. Iranian Math. Soc. 36 (2010), no. 1, 1–25. MR 2743385
- 19.
J.L. Bona, J. Cohen, and G. Wang.
Global well-posedness for a system of KdV-type with coupled quadratic nonlinearities.
Submitted. - 20. Jerry L. Bona, Thierry Colin, and David Lannes, Long wave approximations for water waves, Arch. Ration. Mech. Anal. 178 (2005), no. 3, 373–410. MR 2196497, https://doi.org/10.1007/s00205-005-0378-1
- 21.
J.L. Bona, V.A. Dougalis, O.A. Karakashian, and W.R. McKinney.
Fully discrete methods with grid refinement for the generalized Korteweg-de Vries equation.
In M. Shearer, editor, Proceedings of the workshop on viscous and numerical approximations of shock waves, N.C. State University, pages 117-124. SIAM, Philadelphia, 1990. - 22. J. L. Bona, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 351 (1995), no. 1695, 107–164. MR 1336983, https://doi.org/10.1098/rsta.1995.0027
- 23. J. L. Bona, W. G. Pritchard, and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Roy. Soc. London Ser. A 302 (1981), no. 1471, 457–510. MR 633485, https://doi.org/10.1098/rsta.1981.0178
- 24. J. L. Bona, W. G. Pritchard, and L. R. Scott, A comparison of solutions of two model equations for long waves, Fluid dynamics in astrophysics and geophysics (Chicago, Ill., 1981), Lectures in Appl. Math., vol. 20, Amer. Math. Soc., Providence, R.I., 1983, pp. 235–267. MR 716887
- 25. Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954
- 26. Hongqiu Chen, Long-period limit of nonlinear dispersive waves: the BBM-equation, Differential Integral Equations 19 (2006), no. 4, 463–480. MR 2215629
- 27. 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, https://doi.org/10.1090/S0025-5718-07-02045-5
- 28. Philippe G. Ciarlet, Metod konechnykh èlementov dlya èllipticheskikh zadach, “Mir”, Moscow, 1980 (Russian). Translated from the English by B. I. Kvasov. MR 608971
- 29. Bernardo Cockburn, George E. Karniadakis, and Chi-Wang Shu (eds.), Discontinuous Galerkin methods, Lecture Notes in Computational Science and Engineering, vol. 11, Springer-Verlag, Berlin, 2000. Theory, computation and applications; Papers from the 1st International Symposium held in Newport, RI, May 24–26, 1999. MR 1842160
- 30. Walter Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations 10 (1985), no. 8, 787–1003. MR 795808, https://doi.org/10.1080/03605308508820396
- 31. K. Dekker and J. G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations, CWI Monographs, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. MR 774402
- 32. 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, https://doi.org/10.1090/S0025-5718-1985-0804927-8
- 33. 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, https://doi.org/10.1098/rsta.1978.0064
- 34. Ben-Yu Guo and Jie Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math. 86 (2000), no. 4, 635–654. MR 1794346, https://doi.org/10.1007/PL00005413
- 35. A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-de Vries equation, SIAM Rev. 14 (1972), 582–643. MR 334675, https://doi.org/10.1137/1014101
- 36. 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, https://doi.org/10.1090/S0025-5718-1990-1035935-4
- 37. Yvan Martel and Frank Merle, Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation, Ann. of Math. (2) 155 (2002), no. 1, 235–280. MR 1888800, https://doi.org/10.2307/3062156
- 38. Frank Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. 14 (2001), no. 3, 555–578. MR 1824989, https://doi.org/10.1090/S0894-0347-01-00369-1
- 39. Tadahiro Oh, Diophantine conditions in global well-posedness for coupled KdV-type systems, Electron. J. Differential Equations (2009), No. 52, 48. MR 2505110
- 40. 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, https://doi.org/10.1016/0021-9991(76)90046-2
- 41. Guido Schneider and C. Eugene Wayne, On the validity of 2D-surface water wave models, GAMM Mitt. Ges. Angew. Math. Mech. 25 (2002), no. 1-2, 127–151. MR 2016828
- 42. Alwyn C. Scott, F. Y. F. Chu, and David W. McLaughlin, The soliton: a new concept in applied science, Proc. IEEE 61 (1973), 1443–1483. MR 0358045
- 43. Thiab R. Taha and Mark J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical, J. Comput. Phys. 55 (1984), no. 2, 192–202. MR 762362, https://doi.org/10.1016/0021-9991(84)90002-0
- 44.
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. - 45.
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. - 46. A. C. Vliegenthart, On finite-difference methods for the Korteweg-de Vries equation, J. Engrg. Math. 5 (1971), 137–155. MR 363153, https://doi.org/10.1007/BF01535405
- 47. 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), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 147–169. Publication No. 33. MR 0658322
- 48. Ragnar Winther, A conservative finite element method for the Korteweg-de Vries equation, Math. Comp. 34 (1980), no. 149, 23–43. MR 551289, https://doi.org/10.1090/S0025-5718-1980-0551289-5
- 49. 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, https://doi.org/10.1016/j.cma.2006.10.043
- 50. 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, https://doi.org/10.1137/S0036142901390378
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Additional Information
J. L. Bona
Affiliation:
Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago, Chicago, Illinois 60607
Email:
bona@math.uic.edu
H. Chen
Affiliation:
Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152
Email:
hchen1@memphis.edu
O. Karakashian
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996
Email:
ohannes@math.utk.edu
Y. Xing
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996 – and – the Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
Email:
xingy@math.utk.edu
DOI:
https://doi.org/10.1090/S0025-5718-2013-02661-0
Keywords:
Discontinuous Galerkin methods,
Korteweg–de Vries equation,
error estimates,
conservation laws
Received by editor(s):
June 7, 2011
Received by editor(s) in revised form:
December 6, 2011
Published electronically:
January 7, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.