Binary and ternary oscillations in a cubic numerical scheme

Physica D: Nonlinear Phenomena

Volumn 120, Issue 3-4, 1998, Pages 287-314

  Hayes, Brian T ^a  

^a DUKE UNIVERSITY   (United States)

Author keywords

Finite difference schemes; Modulation equations; Numerical oscillations

Indexed keywords

EID: 0347105718     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(98)00090-6     Document Type: Article

References (23)

1. A.M. Bloch, Y. Kodama, Dispersive regularization of the Whitham equation for the Toda lattice, SIAM J. Appl. Math. 52 (4) (1992) 909-928.
2. H. Flaschka, The Toda lattice, I. Existence of integrals, Phys. Rev. B 9 (1973) 1924-2005.
3. J. Goodman, P.D. Lax, On dispersive difference schemes I, Comm. Pure Appl. Math. 41 (1988) 1-25.
4. J.M. Greenberg, The shock generation problem for a discrete gas with short range repulsive force, Comm. Pure Appl. Math. 45 (1992) 1125-1139.
5. B.T. Hayes, Binary modulated oscillations in a semi-discrete version of burgers equation, Physica D 106 (1997) 287-313.
6. B.T. Hayes, P.G. LeFloch, Nonclassical shock waves in scalar conservation laws, Arch. Rational Mech. Anal. 139 (1997) 1-56.
7. B.L. Holian, H. Flaschka, D.W. McLaughlin, Shock waves in the Toda lattice: Analysis, Phys. Rev. A 24 (1981) 2595-2623.
8. B.L. Holian, G.K. Straub, Molecular dynamics of shock waves in one-dimensional chains, Phys. Rev. B 18 (1978) 1593-1608.
9. T.Y. Hou, P.D. Lax, Dispersive approximations in fluid dynamics, Comm. Pure Appl. Math. 44 (1991) 1-40.
10. M. Kac, P. van Moerbeke, On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices, Adv. Math. 16 (1975) 1-40.
11. P.D. Lax, C.D. Levermore, The small dispersion limit of the Korteweg-de Vries equation I, II and III, Comm. Pure Appl. Math. 36 (1983) 253-290;
12. Comm. Pure Appl. Math. 36 (1983) 571-593;
13. Comm. Pure Appl. Math. 36 (1983) 809-829.
14. P.D. Lax, C.D. Levermore, S. Venakides, The generation and propagation of oscillations in dispersive IVPs and their limiting behavior, in: T. Fokas, V.E. Zakharov (Eds.), Important Developments in Soliton Theory 1980-1990, Springer, Berlin, 1992.
15. R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, Basel, 1990.
16. C.D. Levermore, J.-G. Liu, Large oscillations arising in a dispersive numerical scheme, Physica D 99 (1996) 191-216.
17. K.D. T-R. McLaughlin, A Continuum Limit of the Toda Lattice, Ph.D. Dissertation, New York University, Graduate School of Arts and Science, August 1994.
18. P.D. Miller, O. Bang, Macroscopic dynamics in quadratically nonlinear lattices, Phys. Rev. E, to appear.
19. P.D. Miller, N.M. Ercolani, C.D. Levermore, Modulation of multiphase waves in the presence of resonance, Physica D 92 (1996) 1-27.
20. R.R. Rosales, C.V. Turner, The small dispersion limit for a non-linear semi-discrete system of equations: Part I, preprint, 1995.
21. M. Toda, Theory of Nonlinear Lattices, Series in Solid-State Sciences, vol. 20, 2nd ed., Springer, New York, 1988.
22. S. Venakides, P. Deift, R. Oba, The Toda shock problem, Comm. Pure Appl. Math. 44 (1991) 1041-1117.
23. G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.