Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation. (English) Zbl 0541.65083

Summary: [For part II see the article reviewed above.]
Various numerical methods are used in order to approximate the Korteweg- de Vries equation, namely: (i) Zabusky-Kruskal scheme, (ii) hopscotch method, (iii) a scheme due to Goda, (iv) a proposed local scheme, (v) a proposed global scheme, (vi) a scheme suggested by Kruskal, (vii) split step Fourier method by Tappert, (viii) an improved split step Fourier method, and (ix) pseudospectral method by Fornberg and Whitham. Comparisons between our proposed scheme, which is developed using notions of the inverse scattering transform, and the other utilized schemes are obtained.


65Z05 Applications to the sciences
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35Q99 Partial differential equations of mathematical physics and other areas of application


Zbl 0541.65082
Full Text: DOI


[1] Korteweg, D. J.; deVries, G., On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Wave, Philos. Mag., 39, 422-443 (1895)
[2] Gardner, C. S.; Morikawa, G. K., Similarity in the Asymptotic Behavior of Collision-Free Hydromagnetic Waves and Water Waves, (Res. Report NYO-9082 (1960), New York Univ., Courant Inst. Math. Sci)
[3] Washimi, H.; Taniuti, T., Propagation of ion-acoustic solitary waves of small amplitudes, Phys. Rev. Lett., 17, 996-998 (1966)
[4] Zabusky, N., A Synergetic Approach to Problems of Nonlinear Wave Propagation and Interaction in Nonlinear Partial Differential Equations, (Ames, W. F. (1967)), 223-258, New York · Zbl 0183.18104
[5] Zabusky, N., Computational synergetics and mathematical innovation, J. Comp. Phys., 43, 195-249 (1981) · Zbl 0489.65043
[6] Kruskal, M. D., Asymptotology in numerical computation: progress and plans on the Fermi-Pasta-Ulam problem, (Proceedings, IBM Scientific Computing Symposium on LargeScale Problems in Physics (1965))
[7] van Wungaarden, L., Linear and Non-Linear Dispersion of Pressure Pulses in Liquid-Bubble Mixtures, (6th Symposium on Naval Hydrodynamics. 6th Symposium on Naval Hydrodynamics, Washington, D. C. (1966))
[8] van Wijngaarden, L., On the equations of motion for mixtures of liquid and gas bubbles, J. Fluid Mech., 33, 465-474 (1968) · Zbl 0187.52202
[9] Zakharov, V. E.; Manakov, S. V.; Novikov, S. P.; Pitayevsky, S. P., Theory of Solitons (1980), Nauk: Nauk Moscow · Zbl 0598.35003
[10] Ablowitz, M.; Segur, H., Solitons and the inverse scattering transform (1981), SIAM: SIAM Philadelphia · Zbl 0472.35002
[11] Zabusky, N. J.; Kruskal, M. D., Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15, 240-243 (1965) · Zbl 1201.35174
[12] Gardner, C.; Greene, J.; Kruskal, M.; Miura, R., Phys. Rev. Lett., 19, 1095 (1967) · Zbl 1061.35520
[13] Vliegenthart, A. C., J. Engrg. Math., 5, 137 (1971) · Zbl 0221.76003
[14] Greig, I. S.; Morris, J. Ll., J. Comp. Phys., 20, 60-84 (1976)
[15] Goda, K., On instability of some finite difference schemes for the Korteweg-deVries equation, J. Phys. Soc. Japan, 1 (1975)
[16] Taha, T., (Ph.D. thesis (1982), Clarkson College: Clarkson College Potsdam, New York)
[17] M. D. Kruskal; M. D. Kruskal
[18] Tappert, F., Lect. Appl. Math. Am. Math. Soc., 15, 215-216 (1974)
[19] Fornberg, B.; Whitham, G. B., Phil. Trans. Roy. Soc., 289, 373 (1978) · Zbl 0384.65049
[20] Fornberg, B., Math. Comp., 27, 45-57 (1973) · Zbl 0258.65092
[21] Richtmyer, R. D.; Morton, K. W., Difference Methods for Initial Value Problems (1967), Wiley-Interscience: Wiley-Interscience New York · Zbl 0155.47502
[22] Cooley, J. W.; Lewis, P. A.W.; Welch, P. D., IEEE Trans. Educ. E-12, 1, 27-34 (1969)
[23] Smith, G. D., Numerical Solution of Partial Differential Equations (1965), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0123.11806
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.