Numerical solution of the Stefan’s problem

Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy


2015, Vol. 1. №3(3)

Numerical solution of the Stefan’s problem

For citation: Borodin S. L. 2015. “Numerical solution of the Stefan’s problem”. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, vol. 1, no 3, pp. 164-175. DOI: 10.21684/2411-7978-2015-1-3-164-175

About the author:

Stanislav L. Borodin, Cand. Sci. (Phys.-Math.), Researcher, Tyumen Branch of the Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences; eLibrary AuthorID, ORCID, Web of Science ResearcherID, Scopus Author ID,


All of the most well-known numerical methods for solving the Stefan’s problem, as well as a new method developed by the author are considered for the purpose of choosing the most efficient of them from a perspective of accuracy and computational speed. A comparison is carried out on the results of solving the problem for the boundary motion of “ice-water” phase transition around the vertical well passing through the thickness of permafrost. The conclusions, which are distributed to other multidimensional and multi-front statements of the Stefan’s problem, are made. The mathematical model, the brief description of the considered numerical methods and the boundaries of their applicability are presented. The comparison shows the advantages and disadvantages of different methods. It is demonstrated that the use of the explicit scheme leads to a marked increase in computation time, the six-point symmetric scheme may have oscillating solution; therefore, the implicit scheme is the most preferred. It is concluded that the most efficient method for one-dimensional and one front Stefan’s problems is the method of catching the front in the grid node using the implicit scheme, and the most efficient method for multi-dimensional and multi-front Stefan’s problems is the enthalpy method using the implicit scheme, which has been developed by the author.


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