Geodesic
The improved boundary conditions at the singular points of coordinate systems for non-stationary boundary value problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 2 (1999) no. 4, pp. 373-384

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In the article, two types of non-stationary boundary value problems in polar, cylindrical and spherical coordinates are considered. These are symmetric problems whose initial data do not depend on angle variables and whatever problems. The difference fourth-order accuracy boundary conditions in poles of coordinate systems are constructed for their application in high-order schemes. The conditions represent special difference analogues of the differential equation in the Cartesian coordinates noted in poles. The modes of realization of boundary conditions in high-order schemes based on a method of an approximate factorization are developed. 
@article{SJVM_1999_2_4_a6,
     author = {V. I. Paasonen},
     title = {The improved boundary conditions at the singular points of coordinate systems for non-stationary boundary value problems},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {373--384},
     publisher = {mathdoc},
     volume = {2},
     number = {4},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_1999_2_4_a6/}
}
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V. I. Paasonen. The improved boundary conditions at the singular points of coordinate systems for non-stationary boundary value problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 2 (1999) no. 4, pp. 373-384. http://geodesic.mathdoc.fr/item/SJVM_1999_2_4_a6/