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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.

MSC:
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
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[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) · JFM 26.0881.02
[2] Gardner, C.S.; Morikawa, G.K., Similarity in the asymptotic behavior of collision-free hydromagnetic waves and water waves, ()
[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, (), 223-258, New York
[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, ()
[7] van Wungaarden, L., Linear and non-linear dispersion of pressure pulses in liquid-bubble mixtures, ()
[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 Moscow · Zbl 0598.35003
[10] Ablowitz, M.; Segur, H., Solitons and the inverse scattering transform, (1981), 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)
[13] Vliegenthart, A.C., J. engrg. math., 5, 137, (1971)
[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., ()
[17] \scM. D. Kruskal, private communication, 1981.
[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)
[20] Fornberg, B., Math. comp., 27, 45-57, (1973)
[21] Richtmyer, R.D.; Morton, K.W., Difference methods for initial value problems, (1967), 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 New York · Zbl 0123.11806
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