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Uniform error bounds of an exponential wave integrator for the Klein–Gordon–Schrödinger system in the nonrelativistic and massless limit regime.

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Title: Uniform error bounds of an exponential wave integrator for the Klein–Gordon–Schrödinger system in the nonrelativistic and massless limit regime.
Authors: Li, Jiyong1 (AUTHOR) ljyong406@163.com, Yang, Minghui (AUTHOR)
Source: Mathematics & Computers in Simulation. Jul2025, Vol. 233, p237-258. 22p.
Subjects: Systems integrators, Oscillations, Wavelengths
Abstract: We propose an exponential wave integrator Fourier pseudo-spectral (EWI-FP) method and establish the uniform error bounds for the Klein–Gordon–Schrödinger system (KGSS) with ɛ ∈ (0 , 1 ]. In the nonrelativistic and massless limit regime (0 < ɛ ≪ 1), the solution of KGSS propagates waves with wavelength O (ɛ) in time and amplitude at O (ɛ α † ) where α † = min { α , β + 1 , 2 } with two parameters α and β. The parameters satisfy α ≥ 0 and β ≥ − 1. In this regime, due to the oscillation in time, it is very difficult to develop efficient schemes and make the corresponding error analysis for KGSS. In this paper, firstly, in order to overcome the difficulty of controlling the nonlinear terms, we transform the KGSS into a system with higher derivative. Then we construct an EWI-FP method and provide the error estimates with two bounds at O (h σ + 2 + min { τ / ɛ 1 − α ∗ , τ 2 / ɛ 2 − α † }) and O (h σ + 2 + τ 2 + ɛ α † ) , respectively, where α ∗ = min { 1 , α , 1 + β } , σ has to do with the smoothness of the solution in space, h is mesh size and τ is time step. From the two error bounds, we obtain the error estimates O (h σ + 2 + τ α † ) for α ≥ 0 and β ≥ − 1. Hence, we get uniform second-order error bounds at O (h σ + 2 + τ 2) in time when α ≥ 2 and β ≥ 1 , and uniformly accurate first-order error estimates for any α ≥ 1 and β ≥ 0. We also get uniformly accurate spatial spectral accuracy for any α ≥ 0 and β ≥ − 1. Our numerical results support our conclusions. [ABSTRACT FROM AUTHOR]
Copyright of Mathematics & Computers in Simulation is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Label: Title
  Group: Ti
  Data: Uniform error bounds of an exponential wave integrator for the Klein–Gordon–Schr&#246;dinger system in the nonrelativistic and massless limit regime.
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  Data: &lt;searchLink fieldCode=&quot;AR&quot; term=&quot;%22Li%2C+Jiyong%22&quot;&gt;Li, Jiyong&lt;/searchLink&gt;&lt;relatesTo&gt;1&lt;/relatesTo&gt; (AUTHOR)&lt;i&gt; ljyong406@163.com&lt;/i&gt;&lt;br /&gt;&lt;searchLink fieldCode=&quot;AR&quot; term=&quot;%22Yang%2C+Minghui%22&quot;&gt;Yang, Minghui&lt;/searchLink&gt; (AUTHOR)
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– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: We propose an exponential wave integrator Fourier pseudo-spectral (EWI-FP) method and establish the uniform error bounds for the Klein–Gordon–Schr&#246;dinger system (KGSS) with ɛ ∈ (0 , 1 ]. In the nonrelativistic and massless limit regime (0 &lt; ɛ ≪ 1), the solution of KGSS propagates waves with wavelength O (ɛ) in time and amplitude at O (ɛ α † ) where α † = min { α , β + 1 , 2 } with two parameters α and β. The parameters satisfy α ≥ 0 and β ≥ − 1. In this regime, due to the oscillation in time, it is very difficult to develop efficient schemes and make the corresponding error analysis for KGSS. In this paper, firstly, in order to overcome the difficulty of controlling the nonlinear terms, we transform the KGSS into a system with higher derivative. Then we construct an EWI-FP method and provide the error estimates with two bounds at O (h σ + 2 + min { τ / ɛ 1 − α ∗ , τ 2 / ɛ 2 − α † }) and O (h σ + 2 + τ 2 + ɛ α † ) , respectively, where α ∗ = min { 1 , α , 1 + β } , σ has to do with the smoothness of the solution in space, h is mesh size and τ is time step. From the two error bounds, we obtain the error estimates O (h σ + 2 + τ α † ) for α ≥ 0 and β ≥ − 1. Hence, we get uniform second-order error bounds at O (h σ + 2 + τ 2) in time when α ≥ 2 and β ≥ 1 , and uniformly accurate first-order error estimates for any α ≥ 1 and β ≥ 0. We also get uniformly accurate spatial spectral accuracy for any α ≥ 0 and β ≥ − 1. Our numerical results support our conclusions. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: &lt;i&gt;Copyright of Mathematics &amp; Computers in Simulation is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder&#39;s express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.&lt;/i&gt; (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1016/j.matcom.2025.01.027
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 22
        StartPage: 237
    Subjects:
      – SubjectFull: Systems integrators
        Type: general
      – SubjectFull: Oscillations
        Type: general
      – SubjectFull: Wavelengths
        Type: general
    Titles:
      – TitleFull: Uniform error bounds of an exponential wave integrator for the Klein–Gordon–Schrödinger system in the nonrelativistic and massless limit regime.
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            NameFull: Li, Jiyong
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          Name:
            NameFull: Yang, Minghui
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          Dates:
            – D: 01
              M: 07
              Text: Jul2025
              Type: published
              Y: 2025
          Identifiers:
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              Value: 233
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            – TitleFull: Mathematics & Computers in Simulation
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