An energy-stable generalized-α method for the Swift–Hohenberg equation

A. F. Sarmiento; L. F.R. Espath; Philippe Antoine Vignal Atherton; L. Dalcin; M. Parsani; Victor Calo

doi:10.1016/j.cam.2017.11.004

An energy-stable generalized-α method for the Swift–Hohenberg equation

A. F. Sarmiento^*, L. F.R. Espath, Philippe Antoine Vignal Atherton, L. Dalcin, M. Parsani, Victor Calo

^*Corresponding author for this work

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

18 Scopus citations

Abstract

We propose a second-order accurate energy-stable time-integration method that controls the evolution of numerical instabilities introducing numerical dissipation in the highest-resolved frequencies. Our algorithm further extends the generalized-α method and provides control over dissipation via the spectral radius. We derive the first and second laws of thermodynamics for the Swift–Hohenberg equation and provide a detailed proof of the unconditional energy stability of our algorithm. Finally, we present numerical results to verify the energy stability and its second-order accuracy in time.

Original language	English (US)
Pages (from-to)	836-851
Number of pages	16
Journal	Journal of Computational and Applied Mathematics
Volume	344
DOIs	https://doi.org/10.1016/j.cam.2017.11.004
State	Published - Dec 15 2018

Bibliographical note

Funding Information:
This publication was made possible in part by the CSIRO Professorial Chair in Computational Geoscience at Curtin University and the Deep Earth Imaging Enterprise Future Science Platforms of the Commonwealth Scientific Industrial Research Organisation, CSIRO, of Australia. Additional support was provided by the European Union’s Horizon 2020 Research and Innovation Program of the Marie Skłodowska-Curie grant agreement No. 644602 and the Curtin Institute for Computation . The J. Tinsley Oden Faculty Fellowship Research Program at the Institute for Computational Engineering and Sciences (ICES) of the University of Texas at Austin has partially supported the visits of VMC to ICES.

Funding Information:
This publication was made possible in part by the CSIRO Professorial Chair in Computational Geoscience at Curtin University and the Deep Earth Imaging Enterprise Future Science Platforms of the Commonwealth Scientific Industrial Research Organisation, CSIRO, of Australia. Additional support was provided by the European Union's Horizon 2020 Research and Innovation Program of the Marie Sk?odowska-Curie grant agreement No. 644602 and the Curtin Institute for Computation. The J. Tinsley Oden Faculty Fellowship Research Program at the Institute for Computational Engineering and Sciences (ICES) of the University of Texas at Austin has partially supported the visits of VMC to ICES.

Publisher Copyright:
© 2017 Elsevier B.V.

Keywords

Energy stability
Numerical dissipation
Numerical instability
Pattern formation
Swift–Hohenberg equation
Time integration

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1016/j.cam.2017.11.004

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title = "An energy-stable generalized-α method for the Swift–Hohenberg equation",

abstract = "We propose a second-order accurate energy-stable time-integration method that controls the evolution of numerical instabilities introducing numerical dissipation in the highest-resolved frequencies. Our algorithm further extends the generalized-α method and provides control over dissipation via the spectral radius. We derive the first and second laws of thermodynamics for the Swift–Hohenberg equation and provide a detailed proof of the unconditional energy stability of our algorithm. Finally, we present numerical results to verify the energy stability and its second-order accuracy in time.",

keywords = "Energy stability, Numerical dissipation, Numerical instability, Pattern formation, Swift–Hohenberg equation, Time integration",

author = "Sarmiento, {A. F.} and Espath, {L. F.R.} and {Vignal Atherton}, {Philippe Antoine} and L. Dalcin and M. Parsani and Victor Calo",

note = "Funding Information: This publication was made possible in part by the CSIRO Professorial Chair in Computational Geoscience at Curtin University and the Deep Earth Imaging Enterprise Future Science Platforms of the Commonwealth Scientific Industrial Research Organisation, CSIRO, of Australia. Additional support was provided by the European Union{\textquoteright}s Horizon 2020 Research and Innovation Program of the Marie Sk{\l}odowska-Curie grant agreement No. 644602 and the Curtin Institute for Computation . The J. Tinsley Oden Faculty Fellowship Research Program at the Institute for Computational Engineering and Sciences (ICES) of the University of Texas at Austin has partially supported the visits of VMC to ICES. Funding Information: This publication was made possible in part by the CSIRO Professorial Chair in Computational Geoscience at Curtin University and the Deep Earth Imaging Enterprise Future Science Platforms of the Commonwealth Scientific Industrial Research Organisation, CSIRO, of Australia. Additional support was provided by the European Union's Horizon 2020 Research and Innovation Program of the Marie Sk?odowska-Curie grant agreement No. 644602 and the Curtin Institute for Computation. The J. Tinsley Oden Faculty Fellowship Research Program at the Institute for Computational Engineering and Sciences (ICES) of the University of Texas at Austin has partially supported the visits of VMC to ICES. Publisher Copyright: {\textcopyright} 2017 Elsevier B.V.",

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doi = "10.1016/j.cam.2017.11.004",

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AU - Dalcin, L.

AU - Parsani, M.

AU - Calo, Victor

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N2 - We propose a second-order accurate energy-stable time-integration method that controls the evolution of numerical instabilities introducing numerical dissipation in the highest-resolved frequencies. Our algorithm further extends the generalized-α method and provides control over dissipation via the spectral radius. We derive the first and second laws of thermodynamics for the Swift–Hohenberg equation and provide a detailed proof of the unconditional energy stability of our algorithm. Finally, we present numerical results to verify the energy stability and its second-order accuracy in time.

AB - We propose a second-order accurate energy-stable time-integration method that controls the evolution of numerical instabilities introducing numerical dissipation in the highest-resolved frequencies. Our algorithm further extends the generalized-α method and provides control over dissipation via the spectral radius. We derive the first and second laws of thermodynamics for the Swift–Hohenberg equation and provide a detailed proof of the unconditional energy stability of our algorithm. Finally, we present numerical results to verify the energy stability and its second-order accuracy in time.

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KW - Numerical dissipation

KW - Numerical instability

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KW - Time integration

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ER -

An energy-stable generalized-α method for the Swift–Hohenberg equation

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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