Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension


Author: R. T. Glassey
Journal: Math. Comp. 58 (1992), 83-102
MSC: Primary 65M12; Secondary 35Q60
DOI: https://doi.org/10.1090/S0025-5718-1992-1106968-6
MathSciNet review: 1106968
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An energy-preserving, linearly implicit finite difference scheme is presented for approximating solutions to the periodic Cauchy problem for the one-dimensional Zakharov system of two nonlinear partial differential equations. First-order convergence estimates are obtained in a standard "energy" norm in terms of the initial errors and the usual discretization errors.


References [Enhancements On Off] (What's this?)

  • [1] H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: Smoothness and approximation, J. Funct. Anal. 79 (1988), 183-210. MR 950090 (89h:35273)
  • [2] M. Delfour, M. Fortin, and G. Payne, Finite difference solution of a nonlinear Schrödinger equation, J. Comput. Phys. 44 (1981), 277-288. MR 645840 (83c:65195)
  • [3] J. Gibbons, S. G. Thornhill, M. J. Wardrop, and D. Ter Harr, On the theory of Langmuir solitons, J. Plasma Phys. 17 (1977), 153-170.
  • [4] R. Glassey and J. Schaeffer, Convergence of a second-order scheme for semilinear hyperbolic equations in $ 2 + 1$ dimensions, Math. Comp. 56 (1991), 87-106. MR 1052095 (91h:65140)
  • [5] G. L. Payne, D. R. Nicholson, and R. M. Downie, Numerical solution of the Zakharov equations, J. Comput. Phys. 50 (1983), 482-498. MR 710406 (84m:82079)
  • [6] J. M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schrödinger equation, Math. Comp. 43 (1984), 21-27. MR 744922 (86c:65098)
  • [7] S. Schochet and M. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys. 106 (1986), 569-580. MR 860310 (87j:35227)
  • [8] W. Strauss and L. Vazquez, Numerical solution of a nonlinear Klein-Gordon equation, J. Comput. Phys. 28 (1978), 271-278. MR 0503140 (58:19970)
  • [9] C. Sulem and P. L. Sulem, Regularity properties for the equations of Langmuir turbulence, C. R. Acad. Sci. Paris Sér. A Math. 289 (1979), 173-176.
  • [10] C. Sulem, P. L. Sulem, and H. Frisch, Tracing complex singularities with spectral methods, J. Comput. Phys. 50 (1983), 138-161. MR 702063 (84m:58088)
  • [11] V. E. Zakharov, Collapse of Langmuir waves, Soviet Phys. JETP 35 (1972), 908-912.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M12, 35Q60

Retrieve articles in all journals with MSC: 65M12, 35Q60


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1992-1106968-6
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society