Uniform Stability of Spectral Nonlinear Galerkin Methods

Yinnian He; Kaitai Li; Chunshan Zhao

doi:10.1002/num.20010

Uniform Stability of Spectral Nonlinear Galerkin Methods

Yinnian He, Kaitai Li, Chunshan Zhao

Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

<div class="line" id="line-5"> This article provides a stability analysis for the backward Euler schemes of time discretization applied to the spatially discrete spectral standard and nonlinear Galerkin approximations of the nonstationary Navier-Stokes equations with some appropriate assumption of the data (λ,  u 0 ,  f ). If the backward Euler scheme with the semi-implicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraint Δ t  ≤ (2/λλ 1 ). Moreover, if the backward Euler scheme with the explicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraints Δ t = O (λ <img src="http://binarystore.wiley.com/store/10.1002/num.20010/asset/equation/tex2gif-stack-1.gif?v=1&s=dccde75b2da2c525a7bb1ce132674dfd266decf3"/> ) and Δ t = O (λ <img src="http://binarystore.wiley.com/store/10.1002/num.20010/asset/equation/tex2gif-stack-2.gif?v=1&s=8d43b497a2b7cf1ebff51149825e62cb8623b3fa"/> ), respectively, where λ <img src="http://binarystore.wiley.com/store/10.1002/num.20010/asset/equation/tex2gif-stack-3.gif?v=1&s=29bf110be796dcadbe8870aa8561bfac1e79e07e"/>  ≤ λ <img src="http://binarystore.wiley.com/store/10.1002/num.20010/asset/equation/tex2gif-stack-4.gif?v=1&s=c100c03b8b805a1a17f3091530f43623cdd569d1"/> , which shows that the restriction on the time step of the spectral nonlinear Galerkin method is less than that of the spectral standard Galerkin method. </div>

Original language	American English
Journal	Numerical Methods for Partial Differential Equations
Volume	20
DOIs	https://doi.org/10.1002/num.20010
State	Published - Sep 2004

Keywords

Spectral Nonlinear Galerkin Methods
Uniform Stability

DC Disciplines

Mathematics

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@article{c1977080929042e2bf8eb006114d579e,

title = "Uniform Stability of Spectral Nonlinear Galerkin Methods",

abstract = " This article provides a stability analysis for the backward Euler schemes of time discretization applied to the spatially discrete spectral standard and nonlinear Galerkin approximations of the nonstationary Navier-Stokes equations with some appropriate assumption of the data (λ,  u 0 ,  f ). If the backward Euler scheme with the semi-implicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraint Δ t  ≤ (2/λλ 1 ). Moreover, if the backward Euler scheme with the explicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraints Δ t = O (λ ) and Δ t = O (λ ), respectively, where λ  ≤ λ , which shows that the restriction on the time step of the spectral nonlinear Galerkin method is less than that of the spectral standard Galerkin method. ",

keywords = "Spectral Nonlinear Galerkin Methods, Uniform Stability",

author = "Yinnian He and Kaitai Li and Chunshan Zhao",

note = "Corresponding author Faculty of Science (State Key Laboratory of Multiphase Flow in Power Engineering), Xi'an Jiaotong University, Xi'an 710049, China Department of Mathematics, Xi'an Jiaotong University, Xi'an 710049, China Search for more papers by this author Faculty of Science (State Key Laboratory of Multiphase Flow in Power Engineering), Xi'an Jiaotong",

year = "2004",

month = sep,

doi = "10.1002/num.20010",

language = "American English",

volume = "20",

journal = "Numerical Methods for Partial Differential Equations",

issn = "0749-159X",

publisher = "John Wiley & Sons Inc.",

}

TY - JOUR

T1 - Uniform Stability of Spectral Nonlinear Galerkin Methods

AU - He, Yinnian

AU - Li, Kaitai

AU - Zhao, Chunshan

N1 - Corresponding author Faculty of Science (State Key Laboratory of Multiphase Flow in Power Engineering), Xi'an Jiaotong University, Xi'an 710049, China Department of Mathematics, Xi'an Jiaotong University, Xi'an 710049, China Search for more papers by this author Faculty of Science (State Key Laboratory of Multiphase Flow in Power Engineering), Xi'an Jiaotong

PY - 2004/9

Y1 - 2004/9

N2 - This article provides a stability analysis for the backward Euler schemes of time discretization applied to the spatially discrete spectral standard and nonlinear Galerkin approximations of the nonstationary Navier-Stokes equations with some appropriate assumption of the data (λ,  u 0 ,  f ). If the backward Euler scheme with the semi-implicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraint Δ t  ≤ (2/λλ 1 ). Moreover, if the backward Euler scheme with the explicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraints Δ t = O (λ ) and Δ t = O (λ ), respectively, where λ  ≤ λ , which shows that the restriction on the time step of the spectral nonlinear Galerkin method is less than that of the spectral standard Galerkin method.

AB - This article provides a stability analysis for the backward Euler schemes of time discretization applied to the spatially discrete spectral standard and nonlinear Galerkin approximations of the nonstationary Navier-Stokes equations with some appropriate assumption of the data (λ,  u 0 ,  f ). If the backward Euler scheme with the semi-implicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraint Δ t  ≤ (2/λλ 1 ). Moreover, if the backward Euler scheme with the explicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraints Δ t = O (λ ) and Δ t = O (λ ), respectively, where λ  ≤ λ , which shows that the restriction on the time step of the spectral nonlinear Galerkin method is less than that of the spectral standard Galerkin method.

KW - Spectral Nonlinear Galerkin Methods

KW - Uniform Stability

UR - https://doi.org/10.1002/num.20010

U2 - 10.1002/num.20010

DO - 10.1002/num.20010

M3 - Article

SN - 0749-159X

VL - 20

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

ER -

Uniform Stability of Spectral Nonlinear Galerkin Methods

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Keywords

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