A WAVELET-GALERKIN METHOD FOR SOLVING STEFAN-PROBLEMS

Stefan problems are nonlinear transient problems which involve a domain whose boundary moves in time. The presence of of moving boundary not only hinders the analytical solution to Stefan problems but also leads a numerical procedure to involve an iterative computation of the time-step for a given advancement of the moving boundary. The coordinate transformation approach proposed by Gupta and Kumar (1980) dissociates the boundary advancement from the size of space mesh and thus eliminates the iteration steps for the numerical solution of a 1-D Stefan problem. In this paper, a Galerkin method along with the coordinate transformation is applied to convert 1-D Stefan problems into initial-value problems. The class of compactly supported orthonormal wavelets developed by Daubechies (1988) will be adopted as the Galerkin bases in the spatial domain. For an exact Galerkin formulation, computational algorithms for exactly evaluating the integrals of wavelet bases and their derivatives are derived. In order to avoid the difficulty of accuracy control associated with the finite-difference methods, it is suggested to solve the resulting initial-value problems by the numerical integration scheme that has accuracy control by automatically adjusting the time step. The method is illustrated with solving the Stefan problem concerning the heat transfer in an ice-water medium. The obtained numerical results are more accurate than these obtained by the finite-difference methods.