Criteria of stability of three-layer difference schemes (Q5955517)

The canonical form of the three-layer difference scheme, i.e \[ By\circ_t + {\tau}^2 R y_{\bar t t} + ay = 0, \tag{1} \] where \(A, B, R\) are linear operators acting in the Euclidean space \(H\), \(y=y^n=y(t_n) \in H, t_n=n\tau, n = 0, 1, 2, \dots, y\circ_t=(y^{n=1}- y^{n-1})/(2\tau), y_{\bar t t}= (y^{n+1}-2y^n +y^{n-1})/{{\tau}^2}, y^0, y^1\) are arbitrary elements of the space \(H\) is considered. The necessary conditions of the stability for the three-layer difference schemes (1) with selfadjoint operators \(A, B\) and \(R\) are obtained.