By Vitushkin A. G.
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This method proved to be a convenient tool for the classification of difference schemes with respect to a number of features, which was demonstrated in a more recent work by Shokin et al. (1985). Using a simple example, we will elucidate the procedure of stability investigation by the differential approximation method. 1) with a = const. > 0. , + .... 102). Neglecting 2 the terms of the order of smallness 0(T ' ) and 0(l? ) for 7, > 1 and 72 > 1, we obtain the following partial differential equation: 7 du a7 + fl du ^ = ah cPu r cPu T ^ - 2 ^ - ,, ( 1 1 0 4 ) N o w express the derivative u„ in terms of the jc-derivatives.
For example, in the case of the difference scheme ( 1 . , KI is the well-known Courant number. The Fourier transform enables one to reduce the stability investigation of difference schemes of the Cauchy problem for equations with constant coefficients to a purely algebraic problem on obtaining the conditions of the boundedness of amplification matrix powers. 97) where \ (n,£) is an eigenvalue of the matrix G(£,/I,T), and c is a constant independent of r , h, and k. J. C. Strikwerda ( 1 9 8 9 ) distinguishes between the two forms of the von Neumann condition: the general von N e u m a n n condition ( 1 .
26) STABILITY ANALYSIS OF DIFFERENCE SCHEMES 19 where k(x, t) is a given sufficiently smooth function of x and f, a n d 0 < c, < k(x, t) < c . 29) 0
Analytic capacity and rational approximations by Vitushkin A. G.