Numerical Tests of a Phase Field Model with Second Order Accuracy

Abstract

Numerical computations are performed for a recently derived phase field model for the interface between two phases. The rigorous results indicate that solutions to this new phase field model should converge more rapidly than traditional ones to solutions of the corresponding sharp interface (free boundary) formulation for sufficiently small values of the approximation parameter $\varepsilon $ representing the thickness of the interfacial region. In particular, the distance between the sharp interface of the limiting model and the zero level set of the phase function in the phase field model is of order $\varepsilon^2$ rather than $\varepsilon$. Numerical computations within a three-dimensional spherically symmetric setting compare the computed solutions of this new model with the known exact solutions for the limiting free boundary problem and confirm the second order accuracy predictions of the theory for sufficiently small $\varepsilon$. The sets of parameters include those of succinonitrile used in dendritic experiments.