On the convergence of difference schemes for a heat-conduction equation with discontinuous coefficients

Abstract

THERE is an extensive literature on difference methods of solving equations of the parabolic type. A considerable proportion of the studies concerns equations with constant coefficients. In a number of papers (see, for example, [l]-[6]) the stability and convergence of schemes with continuous and sufficiently smooth coefficients are studied. For example, in [4] the convergence and stability in the mean are proved (see Q 1, Section 2), while uniform stability and convergence are proved in [l]-[3] and [6]. Extension to the case of discontinuous coefficients entails major difficulties, since in the vicinity of the discontinuity the difference scheme does not in general approximate the differential operator [7]. It is only possible to overcome these difficulties for the heat-conduction equation if use is made of the special a priori estimates obtained in [8]. In [7], [9] and [lo] the concept is introduced of homogeneous difference schemes having one and the same computational algorithm at all points of the difference mesh for any coefficients of a differential equation drawn from some class of functions. In this paper we consider homogeneous through-computation schemes for solving linear equations of the parabolic type with discontinuous coefficients without separating out explicitly of the lines of discontinuity-more accurately, without any modifications of the scheme near the lines of discontinuity of the coefficients. Our attention is therefore mainly turned to the question of the convergence of through-computation schemes in the class of discontinuous coefficients. This question was studied for a quasilinear equation in [l 11, where proof was given of the convergence of the scheme Fiz) (see 0 1, Section 3) for the case of moving (“oblique”) discontinuities of the heat-conduction coefficient, on the assumption that h2/7 + 0 when h + 0 and T --f 0 (See also [12]). In this paper this assumption is copied for the linear heat-conduction equation. We shall briefly describe the contents of the present paper. In 0 1 we introduce the initial family of homogeneous difference schemes p$,$ and formulate the mixed difference problem.