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A difference method for the solution of the two-phase Stefan problemстатья
Информация о цитировании статьи получена из
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Дата последнего поиска статьи во внешних источниках: 28 мая 2015 г.
- Авторы: Vasil’ev F.P., Uspenskii A.B.
- Журнал: USSR Computational Mathematics and Mathematical Physics
- Том: 3
- Номер: 5
- Год издания: 1963
- Первая страница: 1192
- Последняя страница: 1208
- DOI: 10.1016/0041-5553(63)90107-1
- Аннотация: In this paper we describe an implicit difference scheme for the solution of the two-phase Stefan problem for a bounded region. We have to deal with such problems in the examination of the heating of bodies with a change in their state of aggregation and are interested in the moving phase interface as well as the body temperature. The condition on the phase interface makes this problem a non-linear one even when the physical characteristics of the medium are piece-wise constant. Of the extensive literature devoted to the Stefan problem we may mention the works [1]-[4] in which the question of the existence of an exact solution is studied, together with the problem of finding an approximate solution. In [1] the Stefan problem is examined in its most general form (the multidimensional case, with an arbitrary number of phase interfaces, when the coefficients in the equation depend on the temperature) and the existence of a generalised solution is proved with the help of an explicit difference scheme. This scheme is not good enough in practice to obtain an approximate solution. In [2] and [3] an approximate solution of the Stefan problem is found by reducing it to a system of ordinary differential equations. In [4] a similar problem is solved by reducing it to integral equations. In [2], [3] and [4] it is assumed that the boundary of the phase interface has been formed and lies strictly inside the body. In [5], [6] there is a description of a difference scheme and the existence of a solution for the more simple (single-phase) problems is proved. In solving the two-phase Stefan problem we use the method developed in [5], [6]. For the numerical solution of the problem we use an implicit difference scheme and give a justification for an iterative method which can be used in conjunction with successive substitution to solve the resulting non-linear difference system (Theorem 1). Then, with certain restrictions on the data of the problem, we prove the convergence of the approximate solution obtained with the implicit difference scheme to the exact solution of the Stefan problem, and so establish the existence of the latter (Theorem 2). © 1963.
- Добавил в систему: Васильев Федор Павлович