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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the instability of leap-frog and Crank-Nicolson approximations of a nonlinear partial differential equation
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by B. Fornberg PDF
Math. Comp. 27 (1973), 45-57 Request permission

Abstract:

It is well known that nonlinear instabilities may occur when the partial differential equations, describing, for example, hydrodynamic flows, are approximated by finite-difference schemes, even if the corresponding linearized equations are stable. A scalar model equation is studied, and it is proved that methods of leap-frog and Crank-Nicolson type are unstable, unless the differential equation is rewritten to make the approximations quasi-conservative. The local structure of the instabilities is discussed.
References
    A. Arakawa, "Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. I," J. Computational Phys., v. 1, 1966, pp. 119-143. B. Fornberg, A Study of the Instability of the Leap-Frog Approximation of a Non-Linear Differential Equation, Report NR 22, June 1969, Department of Computer Sciences, Uppsala University.
  • Heinz-Otto Kreiss and Joseph Oliger, Comparison of accurate methods for the integration of hyperbolic equations, Tellus 24 (1972), 199–215 (English, with Russian summary). MR 319382, DOI 10.3402/tellusa.v24i3.10634
    • N. A. Phillips, "An example of non-linear computational instability," The Atmosphere and the Sea in Motion, Edited by B. Bolin, 1959, Rockefeller Institute, New York, pp. 501-504. R. D. Richtmyer, A Survey of Difference Methods for Non-Steady Fluid Dynamics, NCAR Technical Note 63-2, National Center for Atmospheric Research, Boulder, Colorado, 1962, pp. 16-19.
  • Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Math. Comp. 27 (1973), 45-57
  • MSC: Primary 65M10
  • DOI: https://doi.org/10.1090/S0025-5718-1973-0395249-2
  • MathSciNet review: 0395249