Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension
HTML articles powered by AMS MathViewer
- by R. T. Glassey PDF
- Math. Comp. 58 (1992), 83-102 Request permission
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
- Hélène Added and Stéphane Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal. 79 (1988), no. 1, 183–210. MR 950090, DOI 10.1016/0022-1236(88)90036-5
- M. Delfour, M. Fortin, and G. Payre, Finite-difference solutions of a nonlinear Schrödinger equation, J. Comput. Phys. 44 (1981), no. 2, 277–288. MR 645840, DOI 10.1016/0021-9991(81)90052-8 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.
- Robert Glassey and Jack Schaeffer, Convergence of a second-order scheme for semilinear hyperbolic equations in $2+1$ dimensions, Math. Comp. 56 (1991), no. 193, 87–106. MR 1052095, DOI 10.1090/S0025-5718-1991-1052095-5
- G. L. Payne, D. R. Nicholson, and R. M. Downie, Numerical solution of the Zakharov equations, J. Comput. Phys. 50 (1983), no. 3, 482–498. MR 710406, DOI 10.1016/0021-9991(83)90107-9
- J. M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schroedinger equation, Math. Comp. 43 (1984), no. 167, 21–27. MR 744922, DOI 10.1090/S0025-5718-1984-0744922-X
- Steven H. Schochet and Michael I. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys. 106 (1986), no. 4, 569–580. MR 860310
- Walter Strauss and Luis Vazquez, Numerical solution of a nonlinear Klein-Gordon equation, J. Comput. Phys. 28 (1978), no. 2, 271–278. MR 503140, DOI 10.1016/0021-9991(78)90038-4 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.
- Catherine Sulem, Pierre-Louis Sulem, and Hélène Frisch, Tracing complex singularities with spectral methods, J. Comput. Phys. 50 (1983), no. 1, 138–161. MR 702063, DOI 10.1016/0021-9991(83)90045-1 V. E. Zakharov, Collapse of Langmuir waves, Soviet Phys. JETP 35 (1972), 908-912.
Additional Information
- © Copyright 1992 American Mathematical Society
- 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