Error Estimate for a Finite Difference Crank-Nicolson-ADI Scheme for a Class of Nolinear Parabolic Isotropic Systems

To understand phase transition processes like solidification, phase field models are frequently employed. These models couple the energy (heat) equation for temperature with a nonlinear parabolic partial differential equation (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 a simplified phase field model described by a system of parabolic p.d.e’s, q(ϕ)ϕt = ∇ · (A(ϕ)∇ϕ) + f (ϕ, u), ut = Δu + [p(ϕ)]t, where ϕ(x, y, t) represents the phase indicator function and u(x, y, t) denotes the temperature. The functions q, p, and f are given scalars, and A is a 2×2 diagonal matrix dependent on 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 both equations the Crank-Nicolson-ADI scheme. We prove a convergence result for the method and show results of numerical experiments verifying its order of accuracy. The isotropic system is numerically solved using Crank-Nicolson-ADI finite difference discretization for both equations. The initial-boundary-value problem is considered with homogeneous Dirichlet boundary conditions for ϕ and u. The paper presents preliminary results on finite difference approximations, establishes the main result, showing that finite difference approximations to u and ϕ converge in the discrete L2 and H1 norms with bounds of order Δt2 + h2, given a stability condition of Δt h ≤ σ. Finally, numerical experiments confirm the convergence orders.