Flux-splitting schemes for parabolic problems
Provides a new numerical approach for parabolic problems, but the results are limited to equations without mixed derivatives, making it incremental for a specific class of problems.
The paper constructs unconditionally stable flux locally one-dimensional schemes of first and second order in time for parabolic equations without mixed derivatives, addressing difficulties in solving such problems with mixed derivatives.
To solve numerically boundary value problems for parabolic equations with mixed derivatives, the construction of difference schemes with prescribed quality faces essential difficulties. In parabolic problems, some possibilities are associated with the transition to a new formulation of the problem, where the fluxes (derivatives with respect to a spatial direction) are treated as unknown quantities. In this case, the original problem is rewritten in the form of a boundary value problem for the system of equations in the fluxes. This work deals with studying schemes with weights for parabolic equations written in the flux coordinates. Unconditionally stable flux locally one-dimensional schemes of the first and second order of approximation in time are constructed for parabolic equations without mixed derivatives. A peculiarity of the system of equations written in flux variables for equations with mixed derivatives is that there do exist coupled terms with time derivatives.