Symplectic Finite Difference Approximations of the Nonlinear Klein-Gordon Equation

Research output: Contribution to journalArticle

Abstract

We analyze three finite difference approximations of the nonlinear Klein-Gordon equation and show that they are directly related to symplectic mappings. Two of the schemes, the Perring-Skyrme and Ablowitz-Kruskal-Ladik, are long established, and the third is a new, higher order accurate scheme. We test the schemes on traveling wave and periodic breather problems over long time intervals and compare their accuracy and computational costs with those of symplectic and nonsymplectic method-of-lines approximations and a nonsymplectic energy conserving method.

Original languageEnglish
Pages (from-to)1742-1760
Number of pages19
JournalSIAM Journal on Numerical Analysis
Volume34
Issue number5
Publication statusPublished - 1997

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Nonlinear Klein-Gordon Equation
Finite Difference Approximation
High-order Schemes
Method of Lines
Breathers
Energy Method
Traveling Wave
Computational Cost
Interval
Approximation

Keywords

  • Finite differences
  • Klein-Gordon equation
  • Symplectic approximations

Cite this

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Symplectic Finite Difference Approximations of the Nonlinear Klein-Gordon Equation. / Duncan, D. B.

In: SIAM Journal on Numerical Analysis, Vol. 34, No. 5, 1997, p. 1742-1760.

Research output: Contribution to journalArticle

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AB - We analyze three finite difference approximations of the nonlinear Klein-Gordon equation and show that they are directly related to symplectic mappings. Two of the schemes, the Perring-Skyrme and Ablowitz-Kruskal-Ladik, are long established, and the third is a new, higher order accurate scheme. We test the schemes on traveling wave and periodic breather problems over long time intervals and compare their accuracy and computational costs with those of symplectic and nonsymplectic method-of-lines approximations and a nonsymplectic energy conserving method.

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