Numerical solution of Stefan problems using automatic differentiation

One-dimensional Stefan problems are described by a parabolic partial-differential equation, along with two boundary conditions on a moving boundary. The moving boundary needs to be determined as part of the solution. In this paper, we develop a simple numerical method for solving such problems using a technique known as automatic differentiation. The method obtains a Taylor series expansion for the solution whose coefficients are computed using recursive formulas derived from the differential equation itself. We illustrate the method using the Stefan problem concerning the heat transfer in an ice-water medium. The computational results obtained by the present method are in excellent agreement with the results reported previously by other researchers.