A Legendre-pseudospectral method for computing travelling waves with corners (slope discontinuities) in one space dimension with application to Whitham's equation family

Journal of Computational Physics

Volumn 189, Issue 1, 2003, Pages 98-110

  Boyd, John P ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Cnoidal waves; Corner waves; Pseudospectral algorithm; Whitham equation

Indexed keywords

ITERATIVE METHODS; MATHEMATICAL OPERATORS; NONLINEAR EQUATIONS; NUMERICAL METHODS; WAVE TRANSMISSION;

CNOIDAL WAVE; CORNER WAVES; LEGENDRE PSEUDOSPECTRAL METHODS; NONLINEAR HYPERBOLIC WAVES; PSEUDO-SPECTRAL ALGORITHMS; SLOPE DISCONTINUITIES; SPACE DIMENSIONS; TRAVELLING WAVES; WAVEBREAKING; WHITHAM EQUATIONS;

BOUNDARY CONDITIONS;

EID: 0038454612     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-9991(03)00203-1     Document Type: Article

References (32)

1. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.
2. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.
3. J.P. Boyd, Equatorial solitary waves, Part IV: Kelvin solitons in a shear flow, Dyn. Atmos. Oceans 8 (1984) 173.
4. J.P. Boyd, Solitons from sine waves: analytical and numerical methods for non-integrable solitary and cnoidal waves, Physica D 21 (1986) 227.
5. J.P. Boyd, Peakons and coshoidal waves: traveling wave solutions of the Camassa-Holm equation, Appl. Math. Comput. 81 (1997) 173.
6. J.P. Boyd, Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics: Generalized Solitons and Hyperasymptotic Perturbation Theory, Kluwer Academic Publishers, Amsterdam, 1998.
7. J.P. Boyd, Chebyshev and Fourier Spectral Methods, second ed., Dover, Mineola, NY, 2001.
8. J.P. Boyd, Ostrovsky and Hunter's generic wave equation for weakly dispersive short waves: analytical and numerical study of the parabolic and paraboloidal travelling waves, Eur. J. Appl. Math. (2003) (submitted).
9. E.G. Broadbent, D.W. Moore, Waves of extreme form on a layer of uniform vorticity, Phys. Fluids 28 (1985) 1561.
10. R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661.
11. R. Camassa, D.D. Holm, J.M. Hyman, A new integrable shallow water equation, in: T.-Y. Wu, J.W. Hutchinson (Eds.), Advances in Applied Mechanics, vol. 31, Academic Press, New York, 1994, pp. 1-33.
12. C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods for Fluid Dynamics, Springer, New York, 1988.
13. G-Y. Chen, J.P. Boyd, Nonlinear wave packets of equatorial Kelvin waves, Geophys. Astrophys. Fluid Dyn. 96 (2002) 357.
14. F. Cooper, J.M. Hyman, A. Khare, Compacton solutions in a class of generalized fifth-order Korteweg-de Vries equations, Phys. Rev. E 64 (2001), art. no. 026608.
15. S. Gabov, On Whitham's equation, Soviet Math. Dokl. 19 (1978) 1225.
16. R.H.J. Grimshaw, J.-M. He, L.A. Ostrovsky, Terminal damping of a solitary wave due to radiation in rotational systems, Stud. Appl. Math. 101 (1998) 197.
17. J.K. Hunter, Numerical solutions of some nonlinear dispersive wave equations, Lectures Appl. Math. 26 (1990) 301.
18. H. Liu, E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws, SIAM J. Math. Anal. 33 (2001) 930.
19. A.J. Majda, R.R. Rosales, E.G. Tabak, C.V. Turner, Interaction of large-scale equatorial waves and dispersion of Kelvin waves through topographic resonances, J. Atmos. Sci. 56 (1999) 4118.
20. P. Naumkin, I. Shishmarev, Nonlinear Nonlocal Equations in the Theory of Waves, Trans. Math. Monograph, vol. 133, American Mathematical Society, Providence, 1994.
21. D.I. Pullin, Contour dynamics methods, Annu. Rev. Fluid Mech. 24 (1992) 89.
22. R.E. Robson, A. Prytz, The discrete ordinate/pseudo-spectral method: review and application from a physicist's perspective, Austral. J. Phys. 46 (1993) 465.
23. R.R. Rosales, E.G. Tabak, C.V. Turner, Resonant triads involving a nondispersive wave, Stud. Appl. Math. 108 (2002) 105.
24. P. Rosenau, On nonanalytic solitary waves formed by a nonlinear dispersion, Phys. Lett. A 230 (1997) 305.
25. R.L. Seliger, A note on the breaking of waves, Proc. R. Soc. Lond. A 303 (1968) 493.
26. M. Shefter, R.R. Rosales, Quasiperiodic solutions in weakly nonlinear gas dynamics. Part I. Numerical results in the inviscid case, Stud. Appl. Math. 103 (1999) 279.
27. G.G. Stokes, On the theory of oscillatory waves, Camb. Trans. 8 (1847) 441-473. Also published in vol. 1 of Mathematical and Physical Papers Cambridge University Press, Cambridge, 1880, pp. 197-229.
28. M. Van Dyke, Perturbation Methods in Fluid Mechanics, second ed., Parabolic Press, Stanford, CA, 1975.
29. J.A.C. Weideman, S.C. Reddy, A MATLAB differentiation matrix suite, ACM Trans. Math. Software 26 (2000) 465.
30. G.B. Whitham, Variational methods and applications to water waves, Proc. R. Soc. Lond. A 299 (1967) 6.
31. G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.
32. A.A. Zaitsev, Whitham stationary waves and their dispersion relation, Sov. Phys. Dokl. 31 (1986) 118.