Exponential numerical methods for one-dimensional one-phase Stefan problems

Three exponential iterative methods for one-dimensional one-phase Stefan problems based on the transformation of the moving boundary problem into a mixed one, the discretization of the time variable, and the piecewise linearization of the resulting two-point boundary-value problem at each time step are proposed. Two of the methods are based on the strong conservation-law form of the governing equation and analytically solve a piecewise advection–diffusion equation, whereas the third exponential technique accounts for transient, advective, and diffusive effects when determining the solution. These exponential methods provide piecewise-analytical (exponential) solutions, which, by imposing continuity conditions, are globally continuous throughout the domain, and one of them provides globally smooth solutions. The methods have been applied to the classical one-phase Stefan problem and solutions in excellent agreement with the exact ones have been obtained for several Stefan numbers. In addition, it is shown that the method that accounts for transient, advective, and diffusive effects preserves the similarity of the analytical solution to Stefan problems, yields a tridiagonal matrix, and exhibits a spatial accuracy of, at least, fourth order. Application of this method to a forced one-phase Stefan problem indicates that it provides solutions in excellent agreement with those obtained by means of explicit finite difference and nodal integral techniques, and that the melting-front location exhibits some oscillations in the initial stages whose amplitude decreases as the Stefan number is decreased and as time increases, but which increases as the amplitude of the forcing temperature is increased. It is also shown that the temperature profiles in the liquid are affected by the amplitude and frequency of the forcing and the Stefan number.