Finite Element Methods for a System of Dispersive Equations

Jerry L. Bona; Hongqiu Chen; Ohannes Karakashian; Michael M. Wise

doi:10.1007/s10915-018-0767-x

Finite Element Methods for a System of Dispersive Equations

Jerry L. Bona, Hongqiu Chen, Ohannes Karakashian, Michael M. Wise

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

4 Scopus citations

Abstract

The present study is concerned with the numerical approximation of periodic solutions of systems of Korteweg–de Vries type, coupled through their nonlinear terms. We construct, analyze and numerically validate two types of schemes that differ in their treatment of the third derivatives appearing in the system. One approach preserves a certain important invariant of the system, up to round-off error, while the other, somewhat more standard method introduces a measure of dissipation. For both methods, we prove convergence of a semi-discrete approximation and highlight differences in the basic assumptions required for each. Numerical experiments are also conducted with the aim of ascertaining the accuracy of the two schemes when integrations are made over long time intervals.

Original language	English (US)
Pages (from-to)	1371-1401
Number of pages	31
Journal	Journal of Scientific Computing
Volume	77
Issue number	3
DOIs	https://doi.org/10.1007/s10915-018-0767-x
State	Published - Jul 4 2018
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2022-06-08
Acknowledgements: The work of OK and MW was partially supported by NSF Grant DMS-1620288. HC and JB are grateful for hospitality and support from UT Knoxville during visits there. HC and JB also acknowledge support and fine working conditions at King Abdullah University of Science and Technology in Saudi Arabia, National Taiwan University’s National Center for Theoretical Sciences and the Ulsan National Institute of Science and Technology in South Korea during parts of the development of this project. HC also acknowledges a visiting professorship at the Université de Paris 12 during the initial stages of the project.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Computational Theory and Mathematics
Theoretical Computer Science
Software
General Engineering

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10.1007/s10915-018-0767-x

Cite this

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title = "Finite Element Methods for a System of Dispersive Equations",

abstract = "The present study is concerned with the numerical approximation of periodic solutions of systems of Korteweg–de Vries type, coupled through their nonlinear terms. We construct, analyze and numerically validate two types of schemes that differ in their treatment of the third derivatives appearing in the system. One approach preserves a certain important invariant of the system, up to round-off error, while the other, somewhat more standard method introduces a measure of dissipation. For both methods, we prove convergence of a semi-discrete approximation and highlight differences in the basic assumptions required for each. Numerical experiments are also conducted with the aim of ascertaining the accuracy of the two schemes when integrations are made over long time intervals.",

author = "Bona, {Jerry L.} and Hongqiu Chen and Ohannes Karakashian and Wise, {Michael M.}",

note = "KAUST Repository Item: Exported on 2022-06-08 Acknowledgements: The work of OK and MW was partially supported by NSF Grant DMS-1620288. HC and JB are grateful for hospitality and support from UT Knoxville during visits there. HC and JB also acknowledge support and fine working conditions at King Abdullah University of Science and Technology in Saudi Arabia, National Taiwan University{\textquoteright}s National Center for Theoretical Sciences and the Ulsan National Institute of Science and Technology in South Korea during parts of the development of this project. HC also acknowledges a visiting professorship at the Universit{\'e} de Paris 12 during the initial stages of the project. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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AU - Chen, Hongqiu

AU - Karakashian, Ohannes

AU - Wise, Michael M.

N1 - KAUST Repository Item: Exported on 2022-06-08 Acknowledgements: The work of OK and MW was partially supported by NSF Grant DMS-1620288. HC and JB are grateful for hospitality and support from UT Knoxville during visits there. HC and JB also acknowledge support and fine working conditions at King Abdullah University of Science and Technology in Saudi Arabia, National Taiwan University’s National Center for Theoretical Sciences and the Ulsan National Institute of Science and Technology in South Korea during parts of the development of this project. HC also acknowledges a visiting professorship at the Université de Paris 12 during the initial stages of the project. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2018/7/4

Y1 - 2018/7/4

N2 - The present study is concerned with the numerical approximation of periodic solutions of systems of Korteweg–de Vries type, coupled through their nonlinear terms. We construct, analyze and numerically validate two types of schemes that differ in their treatment of the third derivatives appearing in the system. One approach preserves a certain important invariant of the system, up to round-off error, while the other, somewhat more standard method introduces a measure of dissipation. For both methods, we prove convergence of a semi-discrete approximation and highlight differences in the basic assumptions required for each. Numerical experiments are also conducted with the aim of ascertaining the accuracy of the two schemes when integrations are made over long time intervals.

AB - The present study is concerned with the numerical approximation of periodic solutions of systems of Korteweg–de Vries type, coupled through their nonlinear terms. We construct, analyze and numerically validate two types of schemes that differ in their treatment of the third derivatives appearing in the system. One approach preserves a certain important invariant of the system, up to round-off error, while the other, somewhat more standard method introduces a measure of dissipation. For both methods, we prove convergence of a semi-discrete approximation and highlight differences in the basic assumptions required for each. Numerical experiments are also conducted with the aim of ascertaining the accuracy of the two schemes when integrations are made over long time intervals.

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JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

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Finite Element Methods for a System of Dispersive Equations

Abstract

Bibliographical note

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