Pseudospectral method for the “good” Boussinesq equation

de Frutos, J.; Ortega, T.; Sanz-Serna, J. M.

doi:10.1090/S0025-5718-1991-1079012-6

Pseudospectral method for the “good” Boussinesq equation
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by J. de Frutos, T. Ortega and J. M. Sanz-Serna PDF

Math. Comp. 57 (1991), 109-122 Request permission

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.

References

Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. MR 917480, DOI 10.1007/978-3-642-84108-8
James W. Cooley and John W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comp. 19 (1965), 297–301. MR 178586, DOI 10.1090/S0025-5718-1965-0178586-1

h-dependent stability thresholds avoid the need for a priori bounds in nonlinear convergence proofs

J. de Frutos and J. M. Sanz-Serna, Split-step spectral schemes for nonlinear Dirac systems, J. Comput. Phys. 83 (1989), no. 2, 407–423. MR 1013060, DOI 10.1016/0021-9991(89)90127-7
Bengt Fornberg, On a Fourier method for the integration of hyperbolic equations, SIAM J. Numer. Anal. 12 (1975), no. 4, 509–528. MR 421096, DOI 10.1137/0712040
David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. MR 0520152
J. C. López Marcos and J. M. Sanz-Serna, A definition of stability for nonlinear problems, Numerical treatment of differential equations (Halle, 1987) Teubner-Texte Math., vol. 104, Teubner, Leipzig, 1988, pp. 216–226. MR 1065183
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. MR 967844, DOI 10.1093/imanum/8.1.71
V. S. Manoranjan, A. R. Mitchell, and J. Ll. Morris, Numerical solutions of the good Boussinesq equation, SIAM J. Sci. Statist. Comput. 5 (1984), no. 4, 946–957. MR 765215, DOI 10.1137/0905065
V. S. Manoranjan, T. Ortega, and J. M. Sanz-Serna, Soliton and antisoliton interactions in the “good” Boussinesq equation, J. Math. Phys. 29 (1988), no. 9, 1964–1968. MR 957220, DOI 10.1063/1.527850
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. MR 1069280, DOI 10.1007/BF01385620
Hans J. Stetter, Analysis of discretization methods for ordinary differential equations, Springer Tracts in Natural Philosophy, Vol. 23, Springer-Verlag, New York-Heidelberg, 1973. MR 0426438
Endre Süli, Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math. 53 (1988), no. 4, 459–483. MR 951325, DOI 10.1007/BF01396329
Eitan Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods, SIAM J. Numer. Anal. 23 (1986), no. 1, 1–10. MR 821902, DOI 10.1137/0723001
Thiab R. Taha and Mark J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical, J. Comput. Phys. 55 (1984), no. 2, 192–202. MR 762362, DOI 10.1016/0021-9991(84)90002-0
Robert G. Voigt, David Gottlieb, and M. Yousuff Hussaini (eds.), Spectral methods for partial differential equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1984. MR 758260
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), no. 3, 485–507. MR 842641, DOI 10.1137/0723033