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Numerical analysis of Fourier pseudospectral methods for the Klein–Gordon equation with smooth potentials

Fourier pseudospectral methods for Klein–Gordon equation

Abstract

In this paper, we provide a comprehensive numerical analysis of Fourier-type pseudo-spectral methods applied to the semilinear Klein–Gordon equation (KGE) with smooth potentials. It is shown explicitly that for KGE potentials of finite regularity, the convergence rate of such schemes is at most algebraic and is controlled solely by the regularity of input data and the associated KGE potential. It turns out that the spectral convergence rate can only occur for KGE models with entire potentials. In the latter case, an explicit quantitative error estimate, characterizing spectral convergence rate, is provided. The paper is concluded with several computational examples supporting our theoretical analysis.

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Notes

  1. The proof of inequality (2.7), as given in [26], deals with the integer values of \(r\ge 1\) and \(0\le s\le r\), the general case follows from the standard interpolation argument.

  2. For \(s\ge 1\) the solution is in fact classical.

  3. I.e. if \(\sup _{z\in {\mathbb {C}}} e^{-\sigma |z|^{-p}}|{\mathcal {V}}(z)|<\infty \).

  4. Note that in this case \({\mathcal {A}}U\) reduces to the matrix vector multiplication with a diagonal matrix.

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Correspondence to Nabendra Parumasur.

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Shindin, S., Parumasur, N. & Lukumon, G. Numerical analysis of Fourier pseudospectral methods for the Klein–Gordon equation with smooth potentials. Afr. Mat. 33, 85 (2022). https://doi.org/10.1007/s13370-022-01021-9

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Keywords

  • Fourier-type pseudo-spectral methods
  • Error analysis
  • Spectral convergence
  • Klein-Gordon equation

Mathematics Subject Classification

  • 2010