Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Finite difference method for generalized Zakharov equations


Authors: Qian Shun Chang, Bo Ling Guo and Hong Jiang
Journal: Math. Comp. 64 (1995), 537-553, S7
MSC: Primary 65M06; Secondary 65M12, 76E25, 76M20
DOI: https://doi.org/10.1090/S0025-5718-1995-1284664-5
MathSciNet review: 1284664
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A conservative difference scheme is presented for the initial-boundary value problem for generalized Zakharov equations. The scheme can be implicit or semiexplicit depending on the choice of a parameter. On the basis of a priori estimates and an inequality about norms, convergence of the difference solution is proved in order $ O({h^2} + {\tau ^2})$, which is better than previous results.


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. 29 (1988), 183-210. MR 950090 (89h:35273)
  • [2] I. Blalynickl-Birdla and J. Mycialski, Gaussons: Solitons of the logarithmic Schrödinger equation, Phys. Scripta 20 (1979), 539-544. MR 544500 (80k:81012)
  • [3] R. T. Bullough, P. M. Jack, P. W. Kitchenside, and R. Saunders, Solitons in laser physics, Phys. Scripta 20 (1979), 364-381. MR 544483 (80k:81202)
  • [4] Q. Chang, Conservative difference scheme for generalized nonlinear Schrödinger equations, Scientia Sinica (Series A) 26 (1983), 687-701. MR 721288 (85a:65139)
  • [5] Q. Chang and H. Jiang, A conservative difference scheme for the Zakharov equations, J. Comput. Phys. (to appear). MR 1284857 (95c:76070)
  • [6] Q. Chang and L. Xu, A numerical method for a system of generalized nonlinear Schrödinger equations, J. Comput. Math. 4 (1986), 191-199. MR 860148 (88e:35157)
  • [7] Q. Chang and G. Wang, Multigrid and adaptive algorithm for solving the nonlinear Schrödinger equations, J. Comput. Phys. 88 (1990), 362-380. MR 1059375 (91e:65145)
  • [8] A. Friedman, Partial differential equations, Holt, New York, 1969. MR 0445088 (56:3433)
  • [9] R. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comp. 58 (1992), 83-102. MR 1106968 (92e:65123)
  • [10] -, Approximate solutions to the Zakharov equations via finite differences, J. Comput. Phys. 100 (1992), 377-383. MR 1167750 (93a:76071)
  • [11] K. Konno and H. Suzuki, Self-focussing of laser beam in nonlinear media, Phys. Scripta 20 (1979), 382-386.
  • [12] Milton Lees, Approximate solution of parabolic equations, J. Soc. Indust. Appl. Math. 7 (1959), 167-183. MR 0110212 (22:1092)
  • [13] J. C. Lopez-Marcos and J. M. Sanz-Serna, Stability and convergence in numerical analysis III: Linear investigation of nonlinear stability, IMA J. Numer. Anal. 8 (1988), 71-84. MR 967844 (90j:65079)
  • [14] A. Menikoff, The existence of unbounded solutions of the KdV equation, Comm. Pure Appl. Math. 25 (1972), 407-432. MR 0303129 (46:2267)
  • [15] 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)
  • [16] T. Ortega and J. M. Sanz-Serna, Nonlinear stability and convergence of finite-difference methods for the "good" Boussinesq equation, Numer. Math. 58 (1990), 215-229. MR 1069280 (92b:65067)
  • [17] 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)
  • [18] W. A. Strauss, Mathematical aspects of classical nonlinear field equations, Lecture Notes in Phys., vol. 98, Springer, Berlin, 1979, pp. 123-149. MR 542271 (80h:35092)
  • [19] 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.
  • [20] V. E. Zakharov, Collapse of Langmuir waves, Soviet Phys. JETP 35 (1972), 908-912.
  • [21] P. K. C. Wang, A class of multidimensional nonlinear Langmuir waves, J. Math. Phys. 19 (1978), 1286.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M06, 65M12, 76E25, 76M20

Retrieve articles in all journals with MSC: 65M06, 65M12, 76E25, 76M20


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1995-1284664-5
Keywords: Difference scheme, Zakharov equation, convergence
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society