Numerical solution for the one-phase Stefan problem by Piecewise constant approximation of the interface

The classical one-phase one-dimensional Stefan problem is numerically solved on rectangles, Rj, of increasing size controlled by the Stefan condition. This approach is based on a scheme introduced by E. Di Benedetto and R. Spigler in 1983. The practical implementation rests on the representation via thermal potentials of the solution uj(x, t) to the heat equation in Rj. The quantity uxj(xj, jΔt) which determines the (j+1)-th rectangle is evaluated analytically by solving explicitly an integral equation. The solution in Rj+1 is then obtained by numerically evaluating a further integral expression. The algorithm is tested by solving two problems whose solution is explicitly known. Convergence, stability and convergence rate as Δx→0, Δt→0 have been tested and plots are shown. © 1990 Springer-Verlag