Efficient thermal field computation in phase-field models

We solve the phase-field equations in two dimensions to simulate crystal growth in the low undercooling regime. The novelty is the use of a fast solver for the free space heat equation to compute the thermal field. This solver is based on the efficient direct evaluation of the integral representation of the solution to the constant coefficient, free space heat equation with a smooth source term. The computational cost and memory requirements of the new solver are reasonable and no artificial boundary conditions are needed. This allows one to solve for the thermal field in a computational domain whose size depends only on the size of the growing crystal and not on the extent of the thermal field, which can result in significant computational savings in the low undercooling regime.