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The use of explicit finite difference schemes for low Stefan number problems with moving interface was largely abandoned because they require small time intervals (large CPU time) to obtain accurate non‐oscillatory solutions. This paper uses these type of schemes for better estimations of the dynamics of the solid—liquid interface. The scheme in which time and radial intervals are constant, uses a local, continuous, time‐dependent radial coordinate to define the instantaneous location of the interface. Taylor series expansions which result in a polynomial fit are used for forward and backward interpolation of temperatures of nodal points in the vicinity of the interface. A distinction is made between the left and right position of the interface relative to the closest nodal point. The algorithm handles accurately and effectively the non‐linearities near the interface thus producing accurate stable solutions with relatively low CPU time. The scheme which obviously may be applied to large Stefan number problems, is also suitable for time dependent boundary conditions as well as temperature dependent physical properties. The results obtained by the scheme were in excellent agreement with ones derived from an approximate analytical solution which is applicable in the low Stefan number range.