A Fourier pseudospectral method for the "good" Boussinesq equation with second-order temporal accuracy
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 ear...
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| Published in: | Numerical methods for partial differential equations Vol. 31; no. 1; pp. 202 - 224 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
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Blackwell Publishing Ltd
01-01-2015
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| Abstract | 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 |
|---|---|
| AbstractList | 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
convergence for the solution and
convergence for the time‐derivative of the solution are obtained in this article, instead of the
convergence for the solution and the
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
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 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 [infin] ( 0 , T * ; H 2 ) convergence for the solution and [infin] ( 0 , T * ; 2 ) convergence for the time-derivative of the solution are obtained in this article, instead of the [infin] ( 0 , T * ; 2 ) convergence for the solution and the [infin] ( 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 [Delta] 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 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 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 [Formulaomitted] convergence for the solution and [Formulaomitted] convergence for the time-derivative of the solution are obtained in this article, instead of the [Formulaomitted] convergence for the solution and the [Formulaomitted] 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 [Formulaomitted] required by the proof in De Frutos, et al., Math Comput 57 (1991), 109-122. copyright 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 202-224, 2015 |
| Author | Feng, Wenqiang Wang, Cheng Cheng, Kelong Gottlieb, Sigal |
| Author_xml | – sequence: 1 givenname: Kelong surname: Cheng fullname: Cheng, Kelong organization: Department of Mathematics, Southwest University of Science and Technology, Sichuan, 621010, Mianyang, People's Republic of China – sequence: 2 givenname: Wenqiang surname: Feng fullname: Feng, Wenqiang organization: Department of Mathematics, University of Tennessee, Tennessee, 37996, Knoxville – sequence: 3 givenname: Sigal surname: Gottlieb fullname: Gottlieb, Sigal organization: Department of Mathematics, University of Massachusetts, Massachusetts, 02747, North Dartmouth – sequence: 4 givenname: Cheng surname: Wang fullname: Wang, Cheng email: cwang1@umassd.edu organization: Department of Mathematics, University of Massachusetts, Massachusetts, 02747, North Dartmouth |
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| SubjectTerms | 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 |
| Title | A Fourier pseudospectral method for the "good" Boussinesq equation with second-order temporal accuracy |
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