Operator Splitting Methods for Generalized Korteweg-De Vries Equations

Journal of Computational Physics

Volumn 153, Issue 1, 1999, Pages 203-222

  Holden, Helge ^a   Karlsen, Kenneth Hvistendahl ^b   Risebro, Nils Henrik ^c  

^a NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

^b UNIVERSITY OF BERGEN   (Norway)

^c UNIVERSITY OF OSLO   (Norway)

Author keywords

Generalized KdV equation; Numerical examples; Operator splitting

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; NONLINEAR EQUATIONS;

CONVERGENCE PROPERTIES; DISPERSIVE EQUATIONS; GENERALIZED KORTEWEG-DE VRIE EQUATION; KORTEWEG-DE VRIES-EQUATION; NONLINEAR CONSERVATION LAW; NUMERICAL EXAMPLE; OPERATOR SPLITTING METHOD; OPERATOR-SPLITTING; WEAK SOLUTION;

KORTEWEG-DE VRIES EQUATION;

EID: 0043256997     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.1999.6273     Document Type: Article

References (39)

1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
2. J. L. Bona, V. A. Dougalis, O. A. Karakashian, and W. McKinney, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 351, 107 (1995).
3. W. Craig, T. Kappeler, and W. Strauss, Gain of regularity of equations KdV type, Ann. Inst. H. Poincaré Anal. Non Linéaire 9, 147 (1992).
4. P. G. Drazin and R. S. Johnson, Solitons: An Introduction (Cambridge Univ. Press, Cambridge, UK, 1989).
5. S. Evje and K. H. Karlsen, Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations, Numer. Math., to appear.
6. B. Garcia-Archilla, A supraconvergent scheme for the Korteweg-de Vries equation, Numer. Math. 61, 291 (1992).
7. J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal. 20, 1388 (1989).
8. J. Ginibre, Y. Tsutsumi, and G. Velo, Uniqueness and uniqueness of solutions for the generalized Korteweg-de Vries equation, Math. Z. 203, 9 (1990).
9. J. Goodman and P. D. Lax, On dispersive difference schemes, I, Comm. Pure Appl. Math. 41, 591 (1988).
10. H. Holden and N. H. Risebro, Conservation laws with a random source, Appl. Math. Optim. 36, 229 (1997).
11. T. Y. Hou and P. D. Lax, Dispersive approximation in fluid dynamics, Comm. Pure Appl. Math. 44, 1 (1991).
12. K. H. Karlsen and K.-A. Lie, An unconditionally stable splitting scheme for a class of nonlinear parabolic equations, IMA Numer. Anal., to appear.
13. K. H. Karlsen and N. H. Risebro, Corrected operator splitting for nonlinear parabolic equations, SIAM J. Numer. Anal., to appear.
14. K. H. Karlsen and N. H. Risebro, An operator splitting method for convection-diffusion equations, Numer. Math. 77, 365 (1997).
15. T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Stud. Appl. Math. 8, 93 (1983).
16. C. Kenig, G. Ponce, and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Math. J. 59, 585 (1989).
17. S. N. Kruzhkov and A. V. Faminskii, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Math. USSR-Sb. 120, 391 (1983).
18. P. D. Lax and C. D. Levermore, The small dispersion limit of the Korteweg-de Vries equation, I, Comm. Pure Appl. Math. 36, 253 (1983).
19. P. D. Lax and C. D. Levermore, The small dispersion limit of the Korteweg-de Vries equation, II, Comm. Pure Appl. Math. 36, 571 (1983).
20. P. D. Lax and C. D. Levermore, The small dispersion limit of the Korteweg-de Vries equation, III, Comm. Pure Appl. Math. 36, 809 (1983).
21. C. D. Levermore, The KdV zero-dispersion limit and densities of Dirichlet spectra, in Recent Advances in Partial Differential Equations, Venice 1996, pp. 187-210 (Am. Math. Soc., Providence, RI, 1998).
22. Yi. A. Li, P. J. Olver, and P. Rosenau, Non-analytic solutions of nonlinear wave models, Technical Report 1591, Institute of Mathematics and Applications, Minnesota, 1998.
23. S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons (Consultants Bureau, New York, 1984).
24. A. R. Osborne, Nonlinear Fourier analysis for the infinite-interval Korteweg-de Vries equation. I. An algorithm for the direct scattering transform, J. Comput. Phys. 94, 284 (1991).
25. A. R. Osborne and E. Segre, The numerical inverse scattering transform for the periodic Korteweg-de Vries equation, Phys. Lett. A 173, 131 (1993).
26. A. Provenzale and A. R. Osborne, Nonlinear Fourier analysis for the infinite-interval Korteweg-de Vries equation. II. Numerical tests of the direct scattering transform, J. Comput. Phys. 94, 314 (1991).
27. A. B. Shamardan, Central difference schemes for nonlinear dispersive waves, Comput. Math. Appl. 19, 9 (1990).
28. C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, edited by A. Quarteroni, Lecture Notes in Mathematics (Springer-Verlag, Berlin, to appear).
29. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory schemes, III, J. Comput. Phys. 77, 439 (1988).
30. E. Tadmor, Super-viscosity and spectral approximations of nonlinear conservation laws, in Numerical Methods for Fluid Dynamics IV, edited by K. W. Morton and J. M. Baines (Clarendon Press, Oxford, 1993), pp. 69-81.
31. T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical, J. Comput. Phys. 55, 192 (1984).
32. T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation, J. Comput. Phys. 55, 203 (1984).
33. T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, nonlinear Korteweg-de Vries equation, J. Comput. Phys. 55, 231 (1984).
34. F. Tappert, Numerical solutions of the Korteweg-de Vries equation and its generalizations by the split-step Fourier method, in Nonlinear Wave Motion, edited by A. C. Newell, Lectures in Appl. Math. (Am. Math. Soc., Providence, 1974), Vol. 15, pp. 215-216.
35. S. Venakides, The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data, Trans. Amer. Math. Soc. 301, 189 (1987).
36. S. Venakides, The Korteweg-de Vries equation with small dispersion: Higher order Lax-Levermore theory, Comm. Pure Appl. Math. 43, 335 (1990).
37. S. Venakides, The small dispersion KdV equation with decaying initial data, in Recent Advances in Partial Differential Equations, Venice 1996 (Am. Math. Soc., Providence, RI, 1998), pp. 369-392.
38. S. B. Wineberg, J. F. McGrath, E. F. Gamble, and C. E. Southwell, Implicit spectral methods for wave propagation problems, J. Comput. Phys. 97, 311 (1991).
39. Yi, Zhou, Uniqueness of weak solutions of the KdV equation, Internat. Math. Res. Notices, No. 6, 271 (1997).