On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation

C. Foias; M. S. Jolly; I. G. Kevrekidis; E. S. Titi

doi:10.1016/0375-9601(94)90926-1

On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation

C. Foias, M. S. Jolly, I. G. Kevrekidis, E. S. Titi

Chemical & Biological Engineering

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

22 Scopus citations

Abstract

We show that two fully discrete nonlinear Galerkin schemes based on explicit approximate inertial manifolds preserve the dissipativity of the Kuramoto-Sivashinsky equation (KSE). The radius of the absorbing ball is shown to be uniform in both the time step and number of modes, so that the result holds in the PDE limit. While the schemes are specifically designed to deal with the difficulty of the linear instability in the KSE, simpler schemes can be derived following this approach for other dissipative nonlinear evolutionary equations, such as the 2D Navier-Stokes equations.

Original language	English (US)
Pages (from-to)	87-96
Number of pages	10
Journal	Physics Letters A
Volume	186
Issue number	1-2
DOIs	https://doi.org/10.1016/0375-9601(94)90926-1
State	Published - Mar 7 1994

All Science Journal Classification (ASJC) codes

General Physics and Astronomy

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10.1016/0375-9601(94)90926-1

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title = "On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation",

abstract = "We show that two fully discrete nonlinear Galerkin schemes based on explicit approximate inertial manifolds preserve the dissipativity of the Kuramoto-Sivashinsky equation (KSE). The radius of the absorbing ball is shown to be uniform in both the time step and number of modes, so that the result holds in the PDE limit. While the schemes are specifically designed to deal with the difficulty of the linear instability in the KSE, simpler schemes can be derived following this approach for other dissipative nonlinear evolutionary equations, such as the 2D Navier-Stokes equations.",

author = "C. Foias and Jolly, {M. S.} and Kevrekidis, {I. G.} and Titi, {E. S.}",

note = "Funding Information: The work of C.F. was supported in part by NSF Grants DMS-8802596 and DMS-9007802, that of M.S.J. also by NSF grant DMS-9007802, that of I.G.IC in part by NSF Grant ECS-9023362 and the David and Lucile Packard Foundation, and that of E.S.T. in part by the AFOSR, NSF Grant DMS-8915672, and by the Graduate Council Fund of the University of California, Irvine.",

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doi = "10.1016/0375-9601(94)90926-1",

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T1 - On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation

AU - Foias, C.

AU - Jolly, M. S.

AU - Kevrekidis, I. G.

AU - Titi, E. S.

N1 - Funding Information: The work of C.F. was supported in part by NSF Grants DMS-8802596 and DMS-9007802, that of M.S.J. also by NSF grant DMS-9007802, that of I.G.IC in part by NSF Grant ECS-9023362 and the David and Lucile Packard Foundation, and that of E.S.T. in part by the AFOSR, NSF Grant DMS-8915672, and by the Graduate Council Fund of the University of California, Irvine.

PY - 1994/3/7

Y1 - 1994/3/7

N2 - We show that two fully discrete nonlinear Galerkin schemes based on explicit approximate inertial manifolds preserve the dissipativity of the Kuramoto-Sivashinsky equation (KSE). The radius of the absorbing ball is shown to be uniform in both the time step and number of modes, so that the result holds in the PDE limit. While the schemes are specifically designed to deal with the difficulty of the linear instability in the KSE, simpler schemes can be derived following this approach for other dissipative nonlinear evolutionary equations, such as the 2D Navier-Stokes equations.

AB - We show that two fully discrete nonlinear Galerkin schemes based on explicit approximate inertial manifolds preserve the dissipativity of the Kuramoto-Sivashinsky equation (KSE). The radius of the absorbing ball is shown to be uniform in both the time step and number of modes, so that the result holds in the PDE limit. While the schemes are specifically designed to deal with the difficulty of the linear instability in the KSE, simpler schemes can be derived following this approach for other dissipative nonlinear evolutionary equations, such as the 2D Navier-Stokes equations.

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On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation

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