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

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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. 
@article{ZVMMF_2007_47_4_a9,
     author = {A. A. Zlotnik},
     title = {On the stability of the $\sigma$-scheme with transparent boundary conditions for parabolic equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {671--692},
     publisher = {mathdoc},
     volume = {47},
     number = {4},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_4_a9/}
}
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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/