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.

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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.

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Correspondence to Ohannes Karakashian.

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Dedicated to Professor Bernardo Cockburn on the occasion of his 60th birthday.

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Bona, J.L., Chen, H., Karakashian, O. et al. Finite Element Methods for a System of Dispersive Equations. J Sci Comput 77, 1371–1401 (2018). https://doi.org/10.1007/s10915-018-0767-x

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Keywords

  • Finite element methods
  • Korteweg–de Vries equation
  • Error estimates
  • Conservation laws

Mathematics Subject Classification

  • 65N12
  • 65N30
  • 35Q35
  • 35Q51
  • 35Q53
  • 35Q86
  • 76B15
  • 76B25