Heat and mass transfer in an area with moving borders

Various fields of science and technology require calculations of heat transfer processes in regions of different shapes, the boundaries of which move with time. These are such tasks as a Stefan problem of ground freezing and ingot crystallization, which can be solved analytically under certain assumptions. Later appeared works on calculating the temperature field in burning fuel grains, thermal decomposition of solid mixtures, melting and ablation of heat-resistant coatings. Mathematical models of such processes include the heat equation, initial and boundary conditions, as well as the conditions of boundary displacement. At the same time, there are tasks associated with the calculation of heat and mass transfer processes, when liquid and gas flow in the regions, the boundaries of which move with time. Mathematical models of such problems include systems of partial differential equations, which can be solved only by numerical methods. This paper proposes an effective numerical method for solving such problems on the basis of the finite difference method, which allows tracking the position of region boundaries when they considerably move on the adaptive computational mesh.