Numerical Computation of Nonlinear Waves

  • Bengt Fornberg
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 75)


Equations that can be described as wave equations arise in a large number of physical situations. Most often, they take a form which includes partial derivatives. In contrast to the case of ordinary differential equations, it tends to be true that each equation involving partial derivatives requires a special solution technique. There are of course some very general approaches, like finite differences, finite elements, spectral decomposition etc. but there are likely to be lots of details that differ from case to case. Program packets which more or less automatically handle a wide range of different problems have so far had little or no impact on wave calculations and I do not think any major changes in that respect are in sight.


Solitary Wave Gravity Wave Bifurcation Diagram Conformal Mapping Nonlinear Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Abe, K., and Inoue, O., 1980, Fourier expansion solution of the Korteweg-de Vries equation, J. Comp. Phys., 34: 202.MathSciNetADSMATHCrossRefGoogle Scholar
  2. Bauer, F., Garabedian, P., Korn, D., and Jameson, A., 1975, “Supercritical Wing Sections II,” Springer-Verlag.Google Scholar
  3. Benjamin, T.B., 1967, Instability of periodic wavetrains in nonlinear dispersive systems, Proc. Roy. Soc. A, 299: 59.ADSCrossRefGoogle Scholar
  4. Chakravarthy, S., and Anderson, D., 1979, Numerical conformal mapping, Math. Comp., 33: 953.MathSciNetMATHCrossRefGoogle Scholar
  5. Chen, B., and Saffman, P.G., 1980, Numerical evidence for the existence of new types of gravity waves of permanent form on deep water, Stud. Appl. Math., 62: 1.MathSciNetMATHGoogle Scholar
  6. Eilbeck, J.C., 1978, Numerical studies of solitons, in: “Proc. Symposium on Nonlinear (Soliton) Structure and Dynamics in Condensed Matter, Oxford 1978,” A.R. Bishop and T. Schneider, ed., Springer-Verlag.Google Scholar
  7. Fornberg, B., 1973, On the instability of leap-frog and Crank-Nicolson approximations of a nonlinear partial differential equation, Math. Comp., 27: 45.MathSciNetMATHCrossRefGoogle Scholar
  8. Fornberg, B., 1975, On a Fourier method for the integration of hyperbolic equations, SIAM J. Numer. Anal., 12: 509.Google Scholar
  9. Fornberg, B., 1980, A numerical method for conformal mappings, SIAM J. Sei, and Stat. Comput., 1: 386.Google Scholar
  10. Fornberg, B., and Whitham, G.B., 1978, A numerical and theoretical study of certain nonlinear wave phenomena, Phil. Trans. Roy. Soc. London A, 289: 373.MathSciNetADSMATHCrossRefGoogle Scholar
  11. Gazdag, J., 1973, Numerical convective schemes based on accurate computation of space derivatives, J. Comp. Phys., 13: 100.ADSMATHCrossRefGoogle Scholar
  12. Greig, I.S., and Morris, L.L., 1976, A hopscotch method for the Korteweg-de Vries equation, J. Comp. Phys., 20: 64.MathSciNetADSMATHCrossRefGoogle Scholar
  13. Gustafsson, B., Kreiss, H.O., and Sundström, A., 1972, Stability theory of difference approximations for mixed initial boundary value problems, II, Math. Comp., 26: 649.MathSciNetMATHCrossRefGoogle Scholar
  14. Gutknecht, M.H., 1979, Solving Theodorsen’s integral equation for conformal maps with the fast Fourier transform. Part I. Theory. Research report 79 - 02, Seminar für angewandte Mathematik, Eidgenössische Technische Hochschule, Zürich.Google Scholar
  15. Gutknecht, M.H., 1979, Solving Theodorsen integral equation for conformal maps with the fast Fourier transform. Part II. Practice. Research report 79 - 04, Seminar für angewandte Mathematik, Eidgenössische Technische Hochschule, Zürich.Google Scholar
  16. Hasselmann, D., 1979, The high wavenumber instabilities of a Stokes wave, J. Fluid Mech., 93: 491.ADSMATHCrossRefGoogle Scholar
  17. Hayes, J.K., Kahaner, D.K., and Kellner, R., 1972, An improved method for numerical conformal mapping, Math. Comp, 26: 327.MathSciNetMATHCrossRefGoogle Scholar
  18. Henrici, P., 1979, Fast Fourier methods in computational complex analysis, SIAM Review, 21: 481.MathSciNetMATHCrossRefGoogle Scholar
  19. Keller, H.B., 1977, Numerical solutions of bifurcation and nonlinear eigenvalue problems, in: “Applications of Bifurcation Theory,” Academic Press, New York.Google Scholar
  20. Levi-Civita, T., 1925, Determination rigoureuse des ondes permanentes d’ampleur finie, Math. Ann., 93: 264.MathSciNetMATHCrossRefGoogle Scholar
  21. Lichtenstein, editor, 1925/28, “Jahrbuch liber die Fortschritte der Mathematak 1921-1922,” Walter de Gruyter & Co.Google Scholar
  22. Longuet-Higgins, M.S., 1978, The instabilities of gravity waves of finite amplitude in deep water, Proc. Roy. Soc. London A, 360: 471.MathSciNetADSMATHCrossRefGoogle Scholar
  23. Longuet-Higgins, M.S., and Cokelet, E.D., 1976, The deformation of steep surface waves on water. I. A numerical method of computation, Proc. Roy. Soc. London A, 350: 1.MathSciNetADSMATHCrossRefGoogle Scholar
  24. Longuet-Higgins, M.S., and Cokelet, E.D., 1978, The deformation of steep surface waves on water. II. Growth of normal-mode instabilities, Proc. Roy. Soc. London A, 364: 1.MathSciNetADSMATHCrossRefGoogle Scholar
  25. Longuet-Higgins, M.S., and Fox, M.J.H., 1978, Theory of the almost-highest wave. Part 2. Matching and analytic extension, J. Fluid Mech., 85: 769.MathSciNetADSMATHCrossRefGoogle Scholar
  26. Michell, J.H., 1893, The highest waves in water, Phil. Mag., 36: 430.MATHGoogle Scholar
  27. Milne-Thompson, L.M., 1960, “Theoretical Hydrodynamics,” 4th ed., MacMillan, New York.Google Scholar
  28. Nekrasov, A.I., 1921, On stationary waves (In Russian), Izv Ivanovo-Vosnosonk. Politehn Inst., 3: 52.Google Scholar
  29. Nekrasov, A.I., 1922, On stationary waves. Part 2. On nonlinear integral equations (In Russian), Izv Ivanovo-Vosnosonk. Politehn Inst., 6: 155.Google Scholar
  30. Saffman, P.G., and Yuen, H.C., 1980a, Bifurcation and symmetry breaking in nonlinear dispersive waves, Phys. Rev. Lett., 44: 1097.ADSCrossRefGoogle Scholar
  31. Saffman, P.G., and Yuen, H.C., 1980b, A new type of three- dimensional deep water gravity wave of permanent form, J. Fluid Mech., 101: 797.MathSciNetADSMATHCrossRefGoogle Scholar
  32. Schwartz, L.W., 1974, Computer extensions and analytic continuation of Stokes1 expansion for gravity waves, J. Fluid Mech., 62: 553.ADSMATHCrossRefGoogle Scholar
  33. Stokes, G.G., 1880, Considerations relative to the greatest height of oscillatory waves which can be propagated without change of form, in: “Math, and Phys. Papers,” Cambridge Univ. Press 1.Google Scholar
  34. Stokes, G.G., 1880, Supplement to a paper on the theory of oscillatory waves, in: “Math, and Phys. Papers,” Cambridge Univ. Press 1.Google Scholar
  35. Struik, D.J., 1926, Determination rigoureuse des ondes irrotationelles périodiques dans un canal a profondeur finie, Math. Ann., 95: 595.MathSciNetMATHCrossRefGoogle Scholar
  36. Symm, G.T., 1966, An integral equation method in conformai mapping, Numer. Math., 9: 250.MathSciNetMATHCrossRefGoogle Scholar
  37. Tappert, F., 1974, Numerical solutions of the Korteweg-de Vries equation and its generalizations by the split-step Fourier method, in: “Lectures in Appl. Math., Amer. Math. Soc. 15.”Google Scholar
  38. Toland, J.F., 1978, On the existence of a wave of greatest height and Stoke?s conjecture, Proc. Roy. Soc. London A, 363: 469.MathSciNetADSMATHCrossRefGoogle Scholar
  39. Vliegenthart, A.C., 1971, On finite-difference methods for the Korteweg-de Vries equation, J. Eng. Math., 5: 137.MathSciNetMATHCrossRefGoogle Scholar
  40. Winther, R.W., to appear, A conservative finite element method for the Korteweg-de Vries equation.Google Scholar
  41. Zabusky, N.J., 1968, Solitons and bound states of the time-dependent Schrodinger equation, Phys. Rev., 168: 124.ADSCrossRefGoogle Scholar
  42. Zabusky, N.J., and Kruskal, M.D., 1965, Interactions of 1solitons* in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15: 240.ADSMATHCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1981

Authors and Affiliations

  • Bengt Fornberg
    • 1
  1. 1.Department of Applied MathematicsCalifornia Institute of TechnologyPasadenaUSA

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