For offshore gas pipelines in the northern seas the computation of the total heat flux from ambient in every cross section of the pipeline involves solutions of various non-stationary heat problems, in particular, the one- and multidimensional single- and two-phase Stefan problems. In this paper, we have developed a novel numerical method with increased accuracy in solving one-dimensional single- and two-phase Stefan problems. Results of a comparative analysis of the various numerical techniques with respect to its effectiveness in solving to two-dimensional heat conduction problems in the absence of axial symmetry is presented. The calculation by chosen effective numerical method of the total heat flux into a gas pipeline buried in soil under different conditions presented.