Analysis of a phase-field model for solidification in the limit of vanishing undercooling

Phaseeld models are attractive for numerical simulation of solidi cation. The derivation ensures that moving boundaries in the sharp interface formulation are replaced by thin regions of width given implicitly by the solution. The need to explicitly track the location of geometrically complex solid-liquid interfaces is eliminated. The sharp interface problem is formally recovered as the di use interface thickness ! 0. Karma and Rappel have presented an asymptotic analysis which allows certain restrictions on to be relaxed by adding a small correction to a parameter in the phaseeld equations [A. Karma and W.-J. Rappel, Phys. Rev. E 53 R3017 (1996)]. As a consequence computation can be performed on smaller grids without loss of accuracy in the interface conditions. This is of especial importance at low under-