Geodesic
Flux-splitting schemes for parabolic equations with mixed derivatives
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 8, pp. 1314-1328

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Difference schemes of required quality are often difficult to construct as applied to boundary value problems for parabolic equations with mixed derivatives. Specifically, difficulties arise in the design of monotone difference schemes and unconditionally stable locally one-dimensional splitting schemes. In parabolic problems, certain opportunities are offered by restating the problem in question so that the quantities to be determined are fluxes (directional derivatives). The original problem is then rewritten as a boundary value one for a system of equations in flux variables. Weighted schemes for parabolic equations in flux coordinates are examined. Unconditionally stable locally one-dimensional flux schemes that are first- and second-order accurate in time are constructed for a parabolic equation without mixed derivatives. A feature of systems in flux variables for equations with mixed derivatives is that the terms with time derivatives are coupled with each other. 
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     author = {P. N. Vabishchevich},
     title = {Flux-splitting schemes for parabolic equations with mixed derivatives},
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     publisher = {mathdoc},
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     number = {8},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a8/}
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P. N. Vabishchevich. Flux-splitting schemes for parabolic equations with mixed derivatives. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 8, pp. 1314-1328. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a8/