On the instability of leap-frog and Crank-Nicolson approximations of a nonlinear partial differential equation

Author:
B. Fornberg

Journal:
Math. Comp. **27** (1973), 45-57

MSC:
Primary 65M10

DOI:
https://doi.org/10.1090/S0025-5718-1973-0395249-2

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 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.

**[1]**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.**[2]**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.**[3]**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)****[4]**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.**[5]**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.**[6]**R. D. Richtmyer & K. W. Morton,*Difference Methods for Initial-Value Problems*, 2nd ed., Interscience Tracts in Pure and Appl. Math., no. 4, Interscience, New York, 1967, pp. 128-130. MR**36**#3515. MR**0220455 (36:3515)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1973-0395249-2

Keywords:
Nonlinear instability,
leap-frog scheme,
Crank-Nicolson scheme

Article copyright:
© Copyright 1973
American Mathematical Society