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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
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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 $ L^2$-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.


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  • 1. L. Abdelouhab, J.L. Bona, M. Felland, and J.-C. Saut.
    Non-local models for nonlinear, dispersive waves.
    Physica D, 40:360-392, 1989. MR 1044731 (91d:58033)
  • 2. K. Abe and O. Inoue.
    Fourier expansion solution of the Korteweg-de Vries equation.
    J. Comp. Phys., 34:202-210, 1980. MR 559996 (81a:65113)
  • 3. M. Abramowitz and I. Stegun, editors.
    Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, volume 55 of Applied Mathematics Series.
    National Bureau of Standards, 1965. MR 167642 (29:4914)
  • 4. R. Adams.
    Sobolev Spaces.
    Academic Press, New York, 1975. MR 0450957 (56:9247)
  • 5. A.A. Alazman, J.P. Albert, J.L. Bona, M. Chen, and J. Wu.
    Comparisons between the BBM-equation and a Boussinesq system.
    Advances Differential Eq., 11:121-166, 2006. MR 2194497 (2007b:35258)
  • 6. J.P. Albert, J.L. Bona, and M. Felland.
    A criterion for the formation of singularities for the generalized Korteweg-de Vries equation.
    Matemática Aplicada e Computacional, 7:3-11, 1988. MR 965674 (90d:35248)
  • 7. M.E. Alexander and J. Ll. Morris.
    Galerkin methods applied to some model equations for nonlinear dispersive waves.
    J. Comp. Phys., 30:428-451, 1979. MR 530003 (80c:76006)
  • 8. J. Angulo, J.L. Bona, F. Linares, and M. Scialom.
    Scaling, stability and singularities for nonlinear dispersive wave equations: The critical case.
    Nonlinearity, 15:759-786, 2002. MR 1901105 (2003k:35203)
  • 9. J. Angulo, J.L. Bona, and M. Scialom.
    Stability of cnoidal waves.
    Advances Differential Eq., 11:1321-1374, 2006. MR 2276856 (2007k:35391)
  • 10. G. Baker, V.A. Dougalis, and O.A. Karakashian.
    Convergence of Galerkin approximations for the Korteweg-de Vries equation.
    Math. Comp., 40:419-433, 1983. MR 689464 (84f:65072)
  • 11. G. Baker, W. Jureidini, and O.A. Karakashian.
    Piecewise solenoidal vector fields and the Stokes problem.
    SIAM J. Num. Anal., 27:1466-1485, 1990. MR 1080332 (91m:65246)
  • 12. T.B. Benjamin.
    The stability of solitary waves.
    Proc. Royal Soc. London, Ser. A, 328:153-183, 1972. MR 0338584 (49:3348)
  • 13. T.B. Benjamin, J.L. Bona, and J.J. Mahony.
    Model equations for long waves in nonlinear dispersive systems.
    Philos. Trans. Royal Soc. London, Ser. A, 272:47-78, 1972. MR 0427868 (55:898)
  • 14. J.L. Bona.
    On the stabilty theory of solitary waves.
    Proc. Royal Soc. London, Ser. A, 349:363-374, 1975. MR 0386438 (52:7292)
  • 15. J.L. Bona.
    Model equations for waves in nonlinear, dispersive systems.
    In Proc. Int. Congress of Mathematicians, Helsinki, 1978, volume 2, pages 887-894. Academia Scientiarum Fennica: Hungary, 1980. MR 562704 (83b:35103)
  • 16. J.L. Bona.
    Convergence of periodic wave trains in the limit of large wavelength.
    Appl. Sci. Research, 37:21-30, 1981. MR 633079 (82k:35090)
  • 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 by two-point boundary-value problems: The BBM-equation.
    Bull. Iranian Math. Soc., 36:1-25, 2010. MR 2743385 (2011g:35344)
  • 19. J.L. Bona, J. Cohen, and G. Wang.
    Global well-posedness for a system of KdV-type with coupled quadratic nonlinearities.
    Submitted.
  • 20. J.L. Bona, T. Colin, and D. Lannes.
    Long wave approximations for water waves.
    Arch. Rational Mech. Anal., 178:373-410, 2003. MR 2196497 (2007a:76012)
  • 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 schemes for the Generalized Korteweg-de Vries equation.
    Philos. Trans. Royal Soc. London, Ser. A, 351:107-164, 1995. MR 1336983 (96d:65141)
  • 23. J.L. Bona, W.G. Pritchard, and L.R. Scott.
    An evaluation of a model equation for water waves.
    Philos. Trans. Royal Soc. London, Ser. A, 302:457-510, 1981. MR 633485 (83a:35088)
  • 24. J.L. Bona, W.G. Pritchard, and L.R. Scott.
    A comparison of solutions of two models for long waves.
    In N. Lebovitz, editor, Lectures in Applied Mathematics, volume 20, pages 235-267. American Mathematical Society, Providence, 1983. MR 716887 (84j:76011)
  • 25. S. Brenner and L.R. Scott.
    The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics.
    Springer-Verlag, New York, third edition, 2002. MR 2373954 (2008m:65001)
  • 26. H. Chen.
    Long-period limit of nonlinear dispersive waves: the BBM-equation.
    Differential and Integral Eq., 19:463-480, 2006. MR 2215629 (2007c:35145)
  • 27. Y. Cheng and C.-W. Shu.
    A discontinuous finite element method for time dependent partial differential equations with higher order derivatives.
    Math. Comp., 77:699-730, 2008. MR 2373176 (2008m:65252)
  • 28. P. Ciarlet.
    The Finite Element Method for Elliptic Problems.
    North Holland, Amsterdam, New York, Oxford, 1980. MR 608971 (82c:65068)
  • 29. B. Cockburn, G.E. Karniadakis, and C.-W. Shu.
    Discontinuous Galerkin methods, Theory, Computation and Applications, volume 11 of Springer Lecture Notes in Computational Science and Engineering.
    Springer-Verlag, Heidelberg, New York, 2000. MR 1842160 (2002b:65004)
  • 30. W.L. Craig.
    An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits.
    Comm. Partial Differential Equations, 10:787-1003, 1987. MR 795808 (87f:35210)
  • 31. K. Dekker and J.G. Verwer.
    Stability of Runge-Kutta methods for stiff nonlinear differential equations.
    North Holland, 1984. MR 774402 (86g:65003)
  • 32. V.A. Dougalis and O.A. Karakashian.
    On some high order accurate fully discrete Galerkin methods for the Kortweg-de Vries equation.
    Math. Comp., 45:329-345, 1985. MR 804927 (86m:65118)
  • 33. B. Fornberg and G.B. Whitham.
    A numerical and theoretical study of certain nonlinear wave phenomena.
    Philos. Trans. Royal Soc. London, Ser. A, 289:373-404, 1978. MR 497916 (80i:35156)
  • 34. B.-Y. Guo and J. Shen.
    Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval.
    Numer. Math., 86:635-654, 2000. MR 1794346 (2001h:65152)
  • 35. A. Jeffrey and T. Kakutani.
    Weak nonlinear dispersive waves: A discussion centered around the Korteweg-de Vries equation.
    SIAM Rev., 14:582-643, 1972. MR 0334675 (48:12993)
  • 36. O.A. Karakashian and W. McKinney.
    On optimal high order in time approximations for the Korteweg-de Vries equation.
    Math. Comp., 55:473-496, 1990. MR 1035935 (92h:65172)
  • 37. Y. Martel and F. Merle.
    Stability of blow-up profile and lower bounds on the blow up rate for the critical generalized KdV equation.
    Annals Math., 155:235-280, 2002. MR 1888800 (2003e:35270)
  • 38. F. Merle.
    Existence of blow-up solutions in the energy space for the critical generalized KdV equation.
    J. American Math. Soc., 14:666-678, 2001. MR 1824989 (2002f:35193)
  • 39. T. Oh.
    Diophantine conditions in global well-posedness for coupled Kdv-type systems.
    Elec. J. Differential Eqns., 2009:1-48, 2009. MR 2505110 (2010d:35319)
  • 40. H. Schamel and K. Elsässer.
    The application of the spectral method to nonlinear wave propagation.
    J. Comp. Phys., 22:501-516, 1976. MR 0449164 (56:7469)
  • 41. G. Schneider and C.E. Wayne.
    On the validity of 2d-surface water wave models.
    GAMM Mitt. Ges. Angew. Math. Mech., 25:127-151, 2002. MR 2016828 (2005a:76022)
  • 42. A.C. Scott, F.Y.F. Chu, and D.W. McLaughlin.
    The soliton: A new concept in applied science.
    Proc. IEEE, 61:1443-1483, 1973. MR 0358045 (50:10510)
  • 43. T. Taha and M. Ablowitz.
    Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation.
    J. Comp. Phys., 55:231-253, 1984. MR 762364 (86e:65128c)
  • 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. of Engrg. Math., 5:137-155, 1971. MR 0363153 (50:15591)
  • 47. L.B. Wahlbin.
    A dissipative Galerkin method for the numerical solution of first order hyperbolic equations.
    In C. de Boor, editor, Mathematical Aspects of Finite Elements in Partial Differential Equations, pages 147-169. Academic Press, New York, 1974. MR 0658322 (58:31929)
  • 48. R. Winther.
    A conservative finite element method for the Korteweg-de Vries equation.
    Math. Comp., 34:23-43, 1980. MR 551289 (81a:65108)
  • 49. Y. Xu and C.-W. Shu.
    Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations.
    Computer Methods in Appl. Mech. and Eng., 196:3805-3822, 2007. MR 2340006 (2009e:65139)
  • 50. J. Yan and C.-W. Shu.
    A local discontinuous Galerkin method for KdV type equations.
    SIAM J. Numer. Anal., 40:769-791, 2002. MR 1921677 (2003e:65181)

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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.

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