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

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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.

**[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**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.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1992-1106968-6

Article copyright:
© Copyright 1992
American Mathematical Society