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Fast Chebyshev-polynomial method for simulating the time evolution of linear dynamical systems.

Y L Loh1, S N Taraskin, S R Elliott

  • 1Trinity College, University of Cambridge, Cambridge CB2 1TQ, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 21, 2001
PubMed
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We developed a fast simulation method for linear dynamical systems using Chebyshev polynomials. This approach avoids calculating eigenvectors and eigenfrequencies, offering high precision for applications like quantum diffusion and wave propagation.

Area of Science:

  • Computational physics
  • Applied mathematics
  • Dynamical systems theory

Background:

  • Simulating linear dynamical systems is crucial for various scientific fields.
  • Existing time-integration methods often face limitations in accuracy and computational cost.
  • Understanding systems with eigenmodes requires efficient and precise simulation techniques.

Purpose of the Study:

  • To introduce a novel, fast, and highly accurate method for simulating the time evolution of linear dynamical systems with eigenmodes.
  • To present a method that bypasses the need for explicit eigenvector and eigenfrequency computation.
  • To demonstrate the broad applicability of the proposed simulation technique.

Main Methods:

  • The method utilizes a Chebyshev polynomial expansion of the operator matrix solution.

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  • It operates in the eigenfrequency domain, avoiding direct eigenvector/eigenfrequency calculation.
  • The technique is designed to overcome limitations of traditional time-integration approaches.

    • Main Results:

    • The developed method achieves high accuracy, approaching machine precision.
    • It offers a significant speed advantage over conventional simulation techniques.
    • The method is demonstrated to be effective for vibrational wave-packet propagation in disordered lattices.

    Conclusions:

    • The Chebyshev polynomial expansion provides an efficient and accurate simulation tool for linear dynamical systems.
    • This method offers a robust alternative to standard time-integration, applicable to diverse areas.
    • The simulation technique holds promise for advancing research in quantum diffusion, classical mechanics, and transport theory.