On the instability of leapfrog and CrankNicolson approximations of a nonlinear partial differential equation
Author:
B. Fornberg
Journal:
Math. Comp. 27 (1973), 4557
MSC:
Primary 65M10
MathSciNet review:
0395249
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Abstract: It is well known that nonlinear instabilities may occur when the partial differential equations, describing, for example, hydrodynamic flows, are approximated by finitedifference schemes, even if the corresponding linearized equations are stable. A scalar model equation is studied, and it is proved that methods of leapfrog and CrankNicolson type are unstable, unless the differential equation is rewritten to make the approximations quasiconservative. The local structure of the instabilities is discussed.
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 A. Arakawa, "Computational design for longterm numerical integration of the equations of fluid motion: Twodimensional incompressible flow. I," J. Computational Phys., v. 1, 1966, pp. 119143.
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 B. Fornberg, A Study of the Instability of the LeapFrog Approximation of a NonLinear Differential Equation, Report NR 22, June 1969, Department of Computer Sciences, Uppsala University.
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 H.O. Kreiss & J. Oliger, Comparison of Accurate Methods For the Integration of Hyperbolic Equations, Report NR 36, October 1971, Department of Computer Sciences, Uppsala University. MR 0319382 (47:7926)
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 N. A. Phillips, "An example of nonlinear computational instability," The Atmosphere and the Sea in Motion, Edited by B. Bolin, 1959, Rockefeller Institute, New York, pp. 501504.
 [5]
 R. D. Richtmyer, A Survey of Difference Methods for NonSteady Fluid Dynamics, NCAR Technical Note 632, National Center for Atmospheric Research, Boulder, Colorado, 1962, pp. 1619.
 [6]
 R. D. Richtmyer & K. W. Morton, Difference Methods for InitialValue Problems, 2nd ed., Interscience Tracts in Pure and Appl. Math., no. 4, Interscience, New York, 1967, pp. 128130. MR 36 #3515. MR 0220455 (36:3515)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197303952492
PII:
S 00255718(1973)03952492
Keywords:
Nonlinear instability,
leapfrog scheme,
CrankNicolson scheme
Article copyright:
© Copyright 1973
American Mathematical Society
