Foundations of Computational Mathematics

, Volume 1, Issue 1, pp 69–100

Stiff Oscillatory Systems, Delta Jumps and White Noise

  • B. Cano
  • A. M. Stuart
  • E. Süli
  • J. O. Warren
Article

Abstract

Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of \(N \gg 1\) harmonic oscillators used as a forcing term for a scalar ODE. In the first case the initial conditions are chosen so that the forcing term approximates a delta function as \(N \to \infty\) and in the second case so that it approximates white noise. In both cases the fastest natural frequency of the oscillators is <e6>OM</e6>(N). The model problems are integrated numerically in the stiff regime where the time-step \(\Delta t\) satisfies \(N \Delta t={\cal O}(1).\) The convergence of the algorithms is studied in this case in the limit N → ∞ and Δt → 0.For the white noise problem both strong and weak convergence are considered. Order reduction phenomena are observed numerically and proved theoretically.

AMS Classification. 34A65, 60H10, 65L70, 82C80. 

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Copyright information

© Society for the Foundation of Computational Mathematics 2000

Authors and Affiliations

  • B. Cano
    • 1
  • A. M. Stuart
    • 2
  • E. Süli
    • 3
  • J. O. Warren
    • 4
  1. 1.Departamento de Matemática Applicada y Computación Facultad de Ciencias Universidad de Valladolid Valladolid, Spain ES
  2. 2.Mathematics Institute University of Warwick Coventry CV4 7AL, UK GB
  3. 3.Oxford University Computing Laboratory Wolfson Building, Parks Road Oxford OX1 3QD, UK GB
  4. 4.Scientific Computing and Computational Mathematics Program Gates 288 Stanford University Stanford, CA 94305-9025, USAUS

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