Construction of adaptive difference schemes for solving heat conduction equations

Background. The problem of heat field propagation holds a central position in a wide range of physical and mathematical problems. Heat conduction equations have many applications in different branches of physics and engineering such as, for instance, geophysics, thermodynamics, diffusion theory etc. In spite of external simplicity of these equations, their solutions have complicated and non-uniform properties, often admit no analytical representation, and the solution process often turns to be labor-consuming. In this context, it is topical to develop quite simple methods of approximation of heat fields and solving the corresponding parabolic equations, enabling the most effective application of modern computing facilities. From this point of view the difference methods are of special interest for researchers. Methods and materials. Represented in this paper, the development of explicit difference schemes for solving a one-dimensional heat equation is based on the belonging of heat fields to special functional classes, which are denoted by $P_{r, \gamma, \alpha} (D, M, M_1, a)$. In an earlier paper, the authors constructed local splines for these classes. The said splines approximate the functions from these classes with optimal precision. The nodes of these local splines are used in the present paper as the nodes of non-uniform mesh in the construction of adaptive difference schemes for solving heat conduction equations. Results. The paper briefly reviews the earlier results concerning the functional classes $P_{r, \gamma, \alpha} (D, M, M_1, a)$ including heat fields, and also constructing the local splines that approximate the functions from these classes with optimal precision. On the basis of these results the authors have described the construction and application of adaptive difference schemes for approximate solution of heat conduction equations id detail. Considering concrete examples the researches have compared approximations of heat fields by local splines with uniform and adaptive meshes and also the solution of the heat equation on the mentioned meshes. The obtained results justify the efficiency of the presented schemes. Conclusions. The authors suggest several stable difference schemes providing better approximation of heat fields under substantively smaller application of computing resources. The results of this paper could be used for numerical simulations of a wide range of temperature measurement problems.