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Asymptotic preserving schemes for the Klein–Gordon equation in the non-relativistic limit regime

Abstract

We consider the Klein–Gordon equation in the non-relativistic limit regime, i.e. the speed of light \(c\) tending to infinity. We construct an asymptotic expansion for the solution with respect to the small parameter depending on the inverse of the square of the speed of light. As the first terms of this asymptotic can easily be simulated our approach allows us to construct numerical algorithms that are robust with respect to the large parameter \(c\) producing high oscillations in the exact solution.

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Acknowledgments

We are grateful to Christian Lubich for his helpful comments, and to Markus Penz for fruitful discussions during the preparation of this work.

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Correspondence to Erwan Faou.

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Faou, E., Schratz, K. Asymptotic preserving schemes for the Klein–Gordon equation in the non-relativistic limit regime. Numer. Math. 126, 441–469 (2014). https://doi.org/10.1007/s00211-013-0567-z

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  • DOI: https://doi.org/10.1007/s00211-013-0567-z

Mathematics Subject Classification (2000)

  • 35C20
  • 65M12
  • 35L05