Second order phase field asymptotics for multi-component systems

Abstract

We derive a phase field model which approximates a sharp interface model for solidification of a multicomponent alloy to second order in the interfacial thickness $\epsilon$. Since in numerical computations for phase field models the spatial grid size has to be smaller than $\epsilon$ the new approach allows for considerably more accurate phase field computations than have been possible so far. In the classical approach of matched asymptotic expansions the equations to lowest order in $\epsilon$ lead to the sharp interface problem. Considering the equations to the next order, a correction problem is derived. It turns out that, when taking a possibly non-constant correction term to a kinetic coefficient in the phase field model into account, the correction problem becomes trivial and the approximation of the sharp interface problem is of second order in $\epsilon$. By numerical experiments, the better approximation property is well supported. The computational effort to obtain an error smaller than a given value is investigated revealing an enormous efficiency gain.