A FAST NUMERICAL METHOD FOR A SIMPLIFIED PHASE FIELD MODEL - SFYRAKIS, CHR. A.; DOUGALIS, V. A. (Mathematical Methods In Scattering Theory And Biomedical Engineering, 2006)
We consider the phase field model consisting of the system of p.d.e' s \begin{array}{l} q(\theta)\phi t = \nabla \cdot (A\nabla\phi) + f(\phi, u)\,,\\[2pt] u_t = \Delta u + [p(\phi)]_t\,, \end{array} where ϕ = ϕ(x,y,t) is the phase indicator function, θ = arctan $\frac{\phi_{y}}{\phi_{x}}$, u = u(x,y,t) is the temperature, q, p, and f are given scalar functions, and A is a 2 × 2 matrix of suitable functions of θ. This system describes the evolution of phase and temperature in a two phase medium, and is posed for t ≥ 0 on a rectangle in the x, y space with appropriate boundary and initial conditions. We solve the system numerically by a finite difference ADI method and show the results of relevant numerical experiments in the semi-anisotropic case, where A = I. In the case of an isotropic medium with q = 1, and A a diagonal matrix of functions of ϕ, we prove an error estimate of second order accuracy in space and time.