A rapidly converging phase field model

We propose a phase fleld model that approximates its limiting sharp interface model (free boundary problem) up to second order in interface thickness. A broad range of double-well potentials can be utilized so long as the dynamical coe-cient in the phase equation is adjusted appropriately. This model thereby assures that computation with particular value of interface thickness ", will difier at most by O("2) from the limiting sharp interface problem. As an illustration, the speed of a traveling wave of the phase fleld model is asymptotically expanded to demonstrate that it difiers from the speed of the traveling wave of the limit problem by O("2). at the interface are given by N and vn, and ((¢¢¢)) + ¡ denotes the difierence in the limits from the two sides of the interface. The classical Stefan model has the appealing mathematical feature that the temperature, T(x;t); determines the phase at each point (x;t): By deflnition, T(x;t) > TE implies the material is liquid at that point (or, more generally, in the phase with the higher internal energy), while T(x;t) < TE means it is solid, while T(x;t) = TE deflnes the interface ¡(t): Thus, the condition that T(x;t) = TE at the interface appears to be mathematically convenient. The mathematical study of the classical Stefan model posed di-cult challenges that