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Finite difference method for generalized Zakharov equations. (English) Zbl 0827.65138

The authors consider generalized Zakharov equations describing Langmuir waves in plasmas. They develop a conservative difference scheme for the numerical solution of the equations, and point out the invariants associated with the scheme. They show that the truncation error for the scheme is \(O(h^2 + \tau^2)\) where \(h\) and \(\tau\) are respectively distance and time steps, thereby improving on previous methods. A discussion is also given of the convergence of the process.

MSC:

65Z05 Applications to the sciences
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q72 Other PDE from mechanics (MSC2000)
82D10 Statistical mechanics of plasmas
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[1] 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. · Zbl 0655.76044
[2] Iwo Białynicki-Birula and Jerzy Mycielski, Gaussons: solitons of the logarithmic Schrödinger equation, Phys. Scripta 20 (1979), no. 3-4, 539 – 544. Special issue on solitons in physics. · Zbl 1063.81528
[3] R. K. Bullough, P. M. Jack, P. W. Kitchenside, and R. Saunders, Solitons in laser physics, Phys. Scripta 20 (1979), no. 3-4, 364 – 381. Special issue on solitons in physics. · Zbl 1063.78526
[4] Qian Shun Chang, Conservative difference scheme for generalized nonlinear Schrödinger equations, Sci. Sinica Ser. A 26 (1983), no. 7, 687 – 701. · Zbl 0519.65078
[5] Qian Shun Chang and Hong Jiang, A conservative difference scheme for the Zakharov equations, J. Comput. Phys. 113 (1994), no. 2, 309 – 319. · Zbl 0807.76050
[6] Qian Shun Chang and Lin Bao Xu, A numerical method for a system of generalized nonlinear Schrödinger equations, J. Comput. Math. 4 (1986), no. 3, 191 – 199. · Zbl 0599.65085
[7] Qian Shun Chang and Guo Bin Wang, Multigrid and adaptive algorithm for solving the nonlinear Schrödinger equation, J. Comput. Phys. 88 (1990), no. 2, 362 – 380. · Zbl 0708.65111
[8] Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. · Zbl 0224.35002
[9] R. T. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comp. 58 (1992), no. 197, 83 – 102. · Zbl 0746.65066
[10] R. T. Glassey, Approximate solutions to the Zakharov equations via finite differences, J. Comput. Phys. 100 (1992), no. 2, 377 – 383. · Zbl 0775.78001
[11] K. Konno and H. Suzuki, Self-focussing of laser beam in nonlinear media, Phys. Scripta 20 (1979), 382-386.
[12] Milton Lees, Approximate solutions of parabolic equations, J. Soc. Indust. Appl. Math. 7 (1959), 167 – 183. · Zbl 0086.32801
[13] J. C. López Marcos and J. M. Sanz-Serna, Stability and convergence in numerical analysis. III. Linear investigation of nonlinear stability, IMA J. Numer. Anal. 8 (1988), no. 1, 71 – 84. · Zbl 0695.65042
[14] A. Menikoff, The existence of unbounded solutions of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 25 (1972), 407 – 432. · Zbl 0226.35079
[15] 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. · Zbl 0518.76122
[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), no. 2, 215 – 229. · Zbl 0749.65082
[17] 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. · Zbl 0639.76054
[18] W. Strauss, Mathematical aspects of classical nonlinear field equations, Nonlinear problems in theoretical physics (Proc. IX G.I.F.T. Internat. Sem. Theoret. Phys., Univ. Zaragoza, Jaca, 1978) Lecture Notes in Phys., vol. 98, Springer, Berlin-New York, 1979, pp. 123 – 149.
[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. · Zbl 0431.35077
[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. · Zbl 0389.76092
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