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A Fourier pseudospectral method for the “good” Boussinesq equation with second‐order temporal accuracy
Author(s) -
Cheng Kelong,
Feng Wenqiang,
Gottlieb Sigal,
Wang Cheng
Publication year - 2015
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21899
Subject(s) - mathematics , convergence (economics) , fourier transform , partial differential equation , pseudospectral optimal control , partial derivative , fourier series , stability (learning theory) , fourier analysis , mathematical analysis , pseudo spectral method , computer science , machine learning , economics , economic growth
In this article, we discuss the nonlinear stability and convergence of a fully discrete Fourier pseudospectral method coupled with a specially designed second‐order time‐stepping for the numerical solution of the “good” Boussinesq equation. Our analysis improves the existing results presented in earlier literature in two ways. First, aℓ ∞ ( 0 , T * ; H 2 ) convergence for the solution andℓ ∞ ( 0 , T * ; ℓ 2 ) convergence for the time‐derivative of the solution are obtained in this article, instead of theℓ ∞ ( 0 , T * ; ℓ 2 ) convergence for the solution and theℓ ∞ ( 0 , T * ; H − 2 ) convergence for the time‐derivative, given in De Frutos, et al., Math Comput 57 (1991), 109–122. In addition, we prove that this method is unconditionally stable and convergent for the time step in terms of the spatial grid size, compared with a severe restriction time step restriction Δ t ≤ C h 2required by the proof in De Frutos, et al., Math Comput 57 (1991), 109–122.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 202–224, 2015