A Fourier pseudospectral method for the "good" Boussinesq equation with second-order temporal accuracy

• Published in: Numerical methods for partial differential equations Vol. 31; no. 1; pp. 202 - 224
• Main Authors: Cheng, Kelong, Feng, Wenqiang, Gottlieb, Sigal, Wang, Cheng
• Format: Journal Article
• Language: English
• Published: New York Blackwell Publishing Ltd 01-01-2015; Wiley Subscription Services, Inc
• Subjects: Accuracy; aliasing error; Boussinesq equations; Constrictions; Convergence; Fourier analysis; fully discrete Fourier pseudospectral method; good Boussinesq equation; Joining; Mathematical models; Stability; stability and convergence; Temporal logic
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.21899

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, a
ℓ
∞
(
0
,
T
*
;
H
2
)
convergence for the solution and
ℓ
∞
(
0
,
T
*
;
ℓ
2
)
convergence for the time‐derivative of the solution are obtained in this article, instead of the
ℓ
∞
(
0
,
T
*
;
ℓ
2
)
convergence for the solution and the
ℓ
∞
(
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
2
required 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