Phase-field method for 2D dendritic growth

The phase field method is used to model the free dendritic growth into undercooled melt in 2D. The pair of two nonlinear reaction-diffusion equations - for the temperature and for the phase-field function - are solved numerically by using the finite element method in space. A second order modification of the Runge-Kutta method with extended region of stability and automatic choice of the time step is used to solve the resulting system of ordinary differential equations in time. Numerical experiments are made to analyze the evolution in time of a spherical seed in the isotropic case and for different kinds of anisotropy. The results for the dimensionless dendritic tip velocity, found in the different cases, are compared with the results of the microscopic solvability theory and with numerical results found by the finite difference method