Abstracts

Russian participants

As it is known, the program for the numerical solution of boundary value problems of heat conductivity by the difference methods depends greatly on the geometry of a domain. One of the approaches to the solution of this problem is the use of the fictitious domain method where the initial boundary value problem for the real domain is replaced by the problem which, in a certain sense, is close to the former but is assigned in a simpler domain. From the physical point of view the heat conductivity coefficient of the medium in the predetermined fictitious domain is to be large, the volume heat capacity being small.
The paper considers the two-dimensional problem of heat conductivity with an allowance for phase transitions. The authors use the through calculation scheme with the smoothing of dis-continuous coefficients of volume heat capacity and heat conductivity by the temperature in the phase transition neighborhood in the differential equation of heat conductivity.
The developed mathematical model of a two-dimensional section of the calculated do-main takes account of the annual, monthly and daily change of air temperature. It also considers the change of the thickness and thermophysical properties of snow cover and the effect of the coefficient of convection heat exchange with air depending on the wind velocity.
The above problem is solved by the method of finite differences with the use of the longi-tudinal-transverse scheme. Calculation of the two-dimensional problem is based on the method of splitting along the space coordinates with the help of the variable directions’ scheme and implicit methods, based on running algorithms, to provide stability of one-dimensional problems obtained. The volume heat capacity obtained is approximated by the area of a cell of a flow rec-tangular mesh. At the same time the heat conductivity coefficient is approximated by the average step of a mesh. While writing these functions near a certain node of a mesh, we assume a 9-point pattern in the nodes of which some thermophysical parameters are assigned. Such approximation of thermophysical parameters has a more physical sense and helps to obtain the more smooth solutions to the problems with phase transition in particular, as well as allows to describe configurations of the bodies of “step”-like- and other forms. The results of the calculations show the good stability of the above approximation.
Comparison of the results obtained for the real and fictitious domains has shown that ap-plication of the fictitious domain method simplifies the programming and algorithm of the com-putational experiment and helps to obtain sufficiently reliable data on thermal processes. Study of these processes in a real domain is quite complicated and takes much more time. Thus, the use of the method of fictitious domains is rather effective for the solution of the heat conductivity problems.

Note. Abstracts are published in author's edition

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