Simplified methods of numerical modeling of Stefan's problem with explicit allocate of phase boundaries

Abstract

With the development of computer technology and progress in the field of modeling physical and chemical processes, methods of increasing the accuracy, as well as simplifying the algorithms and methods for calculating mathematical models, become especially relevant. This work is devoted to the Stefan problem, to which the problems of heat transfer with a liquid-solid phase transition and diffusion mass transfer with phase transformations in a solid (decomposition of solid solutions, deposition of diffusion protective coatings) are reduced. The features of numerical modeling of the Stefan problem in multiphase systems are considered. The possibilities and disadvantages of existing numerical methods for solving this problem are analyzed. Three new methods are proposed with the allocate of moving interphase boundaries, on which the grid function suffers a discontinuity of the first kind, which greatly simplify the algorithm for the numerical solution of this problem. The comparison of the proposed algorithms with each other and with existing numerical methods has been carried out on a model problem of reaction diffusion in a solid, which is the Stefan problem in multiphase systems by using boundary and initial conditions that allow its analytical solution. The simulation of two existing and three proposed methods shows that a) the equilibrium concentration method leads to significant errors in the early stages of the diffusion process. But over time, the general properties of diffusion processes lead to a reduction of these errors. Therefore, this method is proposed to calculate the final state of long-term diffusion processes; b) methods of linear interpolation and conserved gradient in practice are not inferior in accuracy to generally accepted methods and can be used to solve the Stefan problems in a multidimensional multiphase setting.