Abstract
When analyzing the stability of difference schemes by the Fourier method one usually investigates the location of zeros some algebraic equation with respect to a unit circle in a complex plane. We propose to investigate instead of this equation the behaviour of a family of curves described by a different algebraic equation whose coefficients are computed on the basis of the coefficients of the above equation and depend on the parameters, the number of which is equal to the number of space variables in the original Cauchy problem for a partial differential equation with constant coefficients. One of the curves of this family describes the behaviour of the stability domain boundary of a difference scheme. It is shown by examples of two well-known difference schemes how the mathematical apparatus of catastrophe theory can be used effectively for the detection of the domains of instability and of stability of a difference scheme. Based on the technique we propose a method for the numerical determination of the stability domain boundary of a difference scheme.
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