Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abe, K., and Inoue, O., 1980, Fourier expansion solution of the Korteweg-de Vries equation, J. Comp. Phys., 34: 202.
Bauer, F., Garabedian, P., Korn, D., and Jameson, A., 1975, “Supercritical Wing Sections II,” Springer-Verlag.
Benjamin, T.B., 1967, Instability of periodic wavetrains in nonlinear dispersive systems, Proc. Roy. Soc. A, 299: 59.
Chakravarthy, S., and Anderson, D., 1979, Numerical conformal mapping, Math. Comp., 33: 953.
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.
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.
Fornberg, B., 1973, On the instability of leap-frog and Crank-Nicolson approximations of a nonlinear partial differential equation, Math. Comp., 27: 45.
Fornberg, B., 1975, On a Fourier method for the integration of hyperbolic equations, SIAM J. Numer. Anal., 12: 509.
Fornberg, B., 1980, A numerical method for conformal mappings, SIAM J. Sei, and Stat. Comput., 1: 386.
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.
Gazdag, J., 1973, Numerical convective schemes based on accurate computation of space derivatives, J. Comp. Phys., 13: 100.
Greig, I.S., and Morris, L.L., 1976, A hopscotch method for the Korteweg-de Vries equation, J. Comp. Phys., 20: 64.
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.
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.
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.
Hasselmann, D., 1979, The high wavenumber instabilities of a Stokes wave, J. Fluid Mech., 93: 491.
Hayes, J.K., Kahaner, D.K., and Kellner, R., 1972, An improved method for numerical conformal mapping, Math. Comp, 26: 327.
Henrici, P., 1979, Fast Fourier methods in computational complex analysis, SIAM Review, 21: 481.
Keller, H.B., 1977, Numerical solutions of bifurcation and nonlinear eigenvalue problems, in: “Applications of Bifurcation Theory,” Academic Press, New York.
Levi-Civita, T., 1925, Determination rigoureuse des ondes permanentes d’ampleur finie, Math. Ann., 93: 264.
Lichtenstein, editor, 1925/28, “Jahrbuch liber die Fortschritte der Mathematak 1921-1922,” Walter de Gruyter & Co.
Longuet-Higgins, M.S., 1978, The instabilities of gravity waves of finite amplitude in deep water, Proc. Roy. Soc. London A, 360: 471.
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.
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.
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.
Michell, J.H., 1893, The highest waves in water, Phil. Mag., 36: 430.
Milne-Thompson, L.M., 1960, “Theoretical Hydrodynamics,” 4th ed., MacMillan, New York.
Nekrasov, A.I., 1921, On stationary waves (In Russian), Izv Ivanovo-Vosnosonk. Politehn Inst., 3: 52.
Nekrasov, A.I., 1922, On stationary waves. Part 2. On nonlinear integral equations (In Russian), Izv Ivanovo-Vosnosonk. Politehn Inst., 6: 155.
Saffman, P.G., and Yuen, H.C., 1980a, Bifurcation and symmetry breaking in nonlinear dispersive waves, Phys. Rev. Lett., 44: 1097.
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.
Schwartz, L.W., 1974, Computer extensions and analytic continuation of Stokes1 expansion for gravity waves, J. Fluid Mech., 62: 553.
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.
Stokes, G.G., 1880, Supplement to a paper on the theory of oscillatory waves, in: “Math, and Phys. Papers,” Cambridge Univ. Press 1.
Struik, D.J., 1926, Determination rigoureuse des ondes irrotationelles périodiques dans un canal a profondeur finie, Math. Ann., 95: 595.
Symm, G.T., 1966, An integral equation method in conformai mapping, Numer. Math., 9: 250.
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.”
Toland, J.F., 1978, On the existence of a wave of greatest height and Stoke?s conjecture, Proc. Roy. Soc. London A, 363: 469.
Vliegenthart, A.C., 1971, On finite-difference methods for the Korteweg-de Vries equation, J. Eng. Math., 5: 137.
Winther, R.W., to appear, A conservative finite element method for the Korteweg-de Vries equation.
Zabusky, N.J., 1968, Solitons and bound states of the time-dependent Schrodinger equation, Phys. Rev., 168: 124.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1981 Plenum Press, New York
About this chapter
Cite this chapter
Fornberg, B. (1981). Numerical Computation of Nonlinear Waves. In: Enns, R.H., Jones, B.L., Miura, R.M., Rangnekar, S.S. (eds) Nonlinear Phenomena in Physics and Biology. NATO Advanced Study Institutes Series, vol 75. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-4106-2_6
Download citation
DOI: https://doi.org/10.1007/978-1-4684-4106-2_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4684-4108-6
Online ISBN: 978-1-4684-4106-2
eBook Packages: Springer Book Archive