Abstract
In order to understand phase transition processes like solidification, phase field models are frequently used. The energy (heat) equation for temperature is coupled with another nonlinear parabolic p.d.e. that includes a second unknown, the phase, which takes characteristic values, such as zero in the solid phase and one in the liquid phase. We consider the parabolic system of p.d.e’ s
q(ϕ)ϕt=∇⋅(A(ϕ)∇ϕ)+f(ϕ,u),
ut=Δu+[p(ϕ)]t,
which may be considered as a simplified phase field model. Here, ϕ=ϕ(x,y,t) is the phase indicator function, u=u(x,y,t) is the temperature, q,p, and f are given scalar functions, and A is a 2×2 diagonal matrix of given functions of ϕ. This system is posed for t≥0 on a rectangle in the x,y plane with appropriate boundary and initial conditions. We solve the system using a finite difference method that uses for first equation recursive Euler and Crank-Nicolson method the other equation. We prove a convergence result for the method and show results of numerical experiments verifying its order of accuracy.
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