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Continuum approach to discreteness.

P G Kevrekidis1, I G Kevrekidis, A R Bishop

  • 1Theoretical Division and Center for Nonlinear Studies, MS B258, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 15, 2002
PubMed
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Continuum models based on Padé approximations effectively model discrete systems, capturing dynamics like oscillations but missing others like Bloch oscillations. These models offer homogenization but have a limited range of validity.

Area of Science:

  • Mathematical modeling
  • Computational physics

Background:

  • Continuum models are essential for simulating complex physical systems.
  • Padé approximations offer a powerful tool for deriving accurate continuum models.

Purpose of the Study:

  • To analytically and numerically investigate continuum models derived from Padé approximations.
  • To assess the effectiveness of these models in representing spatially discrete systems.
  • To determine the limitations and range of validity of these continuum approximations.

Main Methods:

  • Analytical investigation of continuum models.
  • Numerical simulations of spatially discrete and continuum systems.
  • Development of numerical methods for solving continuum equations.
  • Comparison with semicontinuum approximations.

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Main Results:

  • Identified temporal dynamics captured by continuum models, including shape oscillations, radiation effects, and trapping.
  • Highlighted dynamics not captured, such as Peierls-Nabarro barriers and Bloch oscillations.
  • Analyzed the homogenization role of these models for discrete and heterogeneous systems.
  • Established the range of validity for continuum approximations through numerical methods.

Conclusions:

  • Padé-approximated continuum models provide valuable insights into discrete system dynamics but have inherent limitations.
  • These models are effective for homogenization but require careful consideration of their applicability.
  • Numerical validation is crucial for defining the boundaries of these approximations.