Difference schemes for the problem of fusing hyperbolic and parabolic equations

Abstract

4, 1998 DIFFERENCE SCHEMES FOR THE PROBLEM OF FUSING HYPERBOLIC AND PARABOLIC EQUATIONS t) A. A. SamarskiY, P. N. Vabishchevich, S. V. LemeshevskiY, and P. P. Matus UDC 519.63 1. Introduction. In mathematical modeling of physical-chemical processes in composite bodies. it is often necessary to use the mathematical models that are based on equations of different type in different parts of the calculation domain. A particular attention in this event is paid to the fusion conditions on the boundaries of subdomains. The questions of unique solvability of boundary value problems for equations of mixed type are intensely discussed in the literature (see, for instance, [1]). Concerning this class of problems, we distinguish boundary value problems for hyperbolic-para- bolic equations whose unique solvability in the class of weak solutions was considered in [2, 3]. In the present article, taking as an example the simplest one-dimensional boundary value problem, we discuss the questions of numerical solution of the problems. We construct a homogeneous difference scheme [4] that belongs to the class of schemes with variable (discontinuous) weight factors [5-7]. We distinguish the classes of unconditionally stable schemes and study the convergence rate of an approximate solution to an exact solution. 2. Statement of the Problem. Consider thefollowinginitial-boundary value problem of fusing hyperbolic and parabolic equations in the rectangle Q =