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Pseudospectral method for the ``good'' Boussinesq equation

Authors: J. de Frutos, T. Ortega and J. M. Sanz-Serna
Journal: Math. Comp. 57 (1991), 109-122
MSC: Primary 65M12; Secondary 65M70
MathSciNet review: 1079012
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Abstract: We prove the nonlinear stability and convergence of a fully discrete, pseudospectral scheme for the "good" Boussinesq equation $ {u_{tt}} = - {u_{xxxx}} + {u_{xx}} + {({u^2})_{xx}}$. Numerical comparisons with finite difference schemes are also reported.

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  • [1] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods in fluid dynamics, Springer-Verlag, New York, 1988. MR 917480 (89m:76004)
  • [2] J. W. Cooley and J. W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comp. 19 (1965), 297-301. MR 0178586 (31:2843)
  • [3] J. de Frutos and J. M. Sanz-Serna, h-dependent stability thresholds avoid the need for a priori bounds in nonlinear convergence proofs, Computational Mathematics III, Proc. Third Internat. Conf. held in Benin City, Nigeria, January 1988 (S. O. Fatunla, ed.), Boole Press, Dublin (to appear).
  • [4] -, Split-step spectral schemes for nonlinear Dirac systems, J. Comput. Phys. 83 (1989), 407-423. MR 1013060 (91f:81021)
  • [5] B. Fornberg, On a Fourier method for the integration of hyperbolic equations, SIAM J. Numer. Anal. 12 (1975), 509-528. MR 0421096 (54:9101)
  • [6] D. Gottlieb and S. A. Orszag, Numerical analysis of spectral methods: Theory and applications, SIAM, Philadelphia, PA, 1977. MR 0520152 (58:24983)
  • [7] J. C. López-Marcos and J. M. Sanz-Serna, A definition of stability for nonlinear problems, Numerical Treatment of Differential Equations, Proc. Fourth Seminar "NUMDIFF-4" held in Halle 1987 (K. Strehmel, ed.), Teubner-Texte zur Mathematik, Leipzig, 1988, pp. 216-226. MR 1065183 (91g:65144)
  • [8] -, Stability and convergence in numerical analysis. III: Linear investigation of nonlinear stability, IMA J. Numer. Anal. 7 (1988) 71-84. MR 967844 (90j:65079)
  • [9] V. S. Manoranjan, A. R. Mitchell, and J. LL. Morris, Numerical solution of the "good" Boussinesq equation, SIAM J. Sci. Statist. Comput. 5 (1984), 946-957. MR 765215 (86d:65125)
  • [10] V. S. Manoranjan, T. Ortega, and J. M. Sanz-Serna, Soliton and anti-soliton interactions in the "good" Boussinesq equation, J. Math. Phys. 29 (1988), 1964-1968. MR 957220 (89m:35196)
  • [11] 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)
  • [12] H. J. Stetter, Analysis of discretization methods for ordinary differential equations, Springer, Berlin, 1973. MR 0426438 (54:14381)
  • [13] E. Suli, Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equation, Numer. Math. 53 (1988), 459-484. MR 951325 (90b:65214)
  • [14] E. Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods, SIAM J. Numer. Anal. 23 (1986), 1-10. MR 821902 (87c:65048)
  • [15] T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equation. II: Numerical nonlinear Schrödinger equation, J. Comput. Phys. 55 (1984), 203-230. MR 762363 (86e:65128b)
  • [16] R. G. Voigt, D. Gottlieb, and M. Y. Hussaini (eds.), Spectral methods for partial differential equations, SIAM, Philadelphia, PA, 1984. MR 758260 (85g:76003)
  • [17] J. A. C. Weideman and B. M. Herbst, Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 23 (1986), 485-507. MR 842641 (87h:65159)

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