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.

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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