The method of approximate fundamental solutions (MAFS) for Stefan problems

The paper presents a new meshless numerical technique for solving one and two-dimensional Stefan problems. The technique presented is based on the use of the delta-shaped functions and the method of approximate fundamental solutions (MAFS) first suggested for solving elliptic problems and heat equations in domains with fixed boundaries. The one-dimensional problems in the plane and cylindrical geometries are considered. The numerical examples are presented and the results are compared with the analytical solutions. The comparison shows that the method presented provides a very high precision in determining the position of the moving boundary even for degenerate and singular problems when a region initially has zero thickness. The same technique was developed for 2D Stefan problems with completely or partially unknown boundary.

► A meshless method is suggested for 1D and 2D plane and cylindrical Stefan problems.
► The regular, degenerate and singular problems are considered.
► The MAFS utilizes truncated Fourier series as approximate fundamental solutions.
► The position of the interface is determined with a very high precision.