Geodesic
On the stability of the $\sigma$-scheme with transparent boundary conditions for parabolic equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 4, pp. 671-692 
A. A. Zlotnik. On the stability of the $\sigma$-scheme with transparent boundary conditions for parabolic equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 4, pp. 671-692. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_4_a9/
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     title = {On the stability of the $\sigma$-scheme with transparent boundary conditions for parabolic equations},
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Voir la notice de l'article provenant de la source Math-Net.Ru

Initial-boundary value problems for self-adjoint parabolic equations on a semiaxis and a semibounded strip are considered. For finite-difference $\sigma$-schemes, an alternative method for stating approximate transparent boundary conditions is suggested and conditions ensuring unconditional stability in the energy norm with respect to the initial data and free terms for a weight $\sigma\ge1/2$ are presented. The validity of these stability conditions in the case of discrete transparent boundary conditions is proved (by several methods), and the derivation of the latter conditions is revisited.

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