Mesh scheme for a phase transition problem with time-fractional derivative

摘要

The time-fractional phase transition problem, formulated in enthalpy form, is studied. This nonlinear problem with an unknown moving boundary includes, as an example, a mathematical model of one-phase Stefan problem with the latent heat accumulation memory. The posed problem is approximated by the backward Euler mesh scheme. The unique solvability of the mesh scheme is proved and a priori estimates for the solution are obtained. The properties of the mesh problem are studied, in particular, an estimate of movement rate for the mesh phase transition boundary is established. The proved estimate make it possible to localize the phase transition boundary and split the mesh scheme into the sum of a nonlinear problem of small algebraic dimension and a larger linear problem. This information can be used for further construction of efficient algorithms for implementing the mesh scheme. Several algorithms for implementing mesh scheme are briefly discussed.