Rapidly converging phase field models via second order asymptotics

We consider phase field models with the objective of approximating sharp interface models. By using second order asymptotics in the interface thickness parameter, $\epsi$, we develop models in which the order $\epsi$ term is eliminated, suggesting more rapid convergence to the $\epsi$ = 0 (sharp interface) limit. In addition we use non-smooth potentials with a non-zero gradient at the roots. These changes result in an error that is 1/200 of the classical models in sample one-dimensional calculations. Alternatively, one can use 1/100 of the node points in each direction for our proposed models and still obtain the same accuracy as one would with the classical model. We expect that this will greatly facilitate two and three dimensional calculations of dendritic growth with physically realistic parameters.