Geodesic
Necessary conditions for the stability of difference schemes
Matematičeskie zametki, Tome 16 (1974) no. 4, pp. 545-552

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For difference equations of the form $u^{n+1}=Au^n+f^nh$, $n\le T/h$ the necessary condition for stability due to von Neumann is well known; this condition is expressed in terms of the spectrum of the operator $A$: $r(A)\le1+ch$. In this note, for a certain class of difference equations, we express this condition in terms of the spectral radius of the symbol of the operator $A$. 
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     author = {V. V. Kucherenko},
     title = {Necessary conditions for the stability of difference schemes},
     journal = {Matemati\v{c}eskie zametki},
     pages = {545--552},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_4_a5/}
}
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V. V. Kucherenko. Necessary conditions for the stability of difference schemes. Matematičeskie zametki, Tome 16 (1974) no. 4, pp. 545-552. http://geodesic.mathdoc.fr/item/MZM_1974_16_4_a5/