Flux-splitting schemes for parabolic problems

TL;DR

This paper develops unconditionally stable flux-splitting difference schemes for parabolic equations, including those with mixed derivatives, by reformulating the problems in flux variables and addressing stability and approximation order.

Contribution

It introduces flux-based schemes for parabolic problems, including mixed derivatives, with stability and accuracy improvements over traditional methods.

Findings

Constructed unconditionally stable first and second order schemes

Successfully handled equations with mixed derivatives

Reformulated problems in flux variables for better stability

Abstract

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.