Local one dimensional difference schemes on non-uniform nets

Abstract

A number of economical difference schemes which are applicable to regions of a special form (parallelepipeds or regions formed from them, cf. [91) have been put forward for the numerical solution of the linear heat conduction equation in several space variables ([II191). Equations with variable coefficients were studied in [~I-[121 and the other works are concerned with constant coefficients. In [lOI we proposed a local one-dimensional method of variable directions for the linear equation (l-l,,) (cf. Section 1) and for the simplest quasi-linear equation (l.ll), and this method was suitable for an arbitrary region in space G, on the boundary r of which boundary conditions of the first kind were given. We constructed a family of local one-dimensional schemes which were homogeneous with respect to space and cyclically homogeneous with respect to time. It was shown that all these schemes were absolutely stable with respect to the initial and the boundary data, andvalso with respect to the right-hand side of the equation (Theorem I, [lo]) and that they converged uniformly at a rate O(h’) + O(T), i.e. had the same order of accuracy as multi-dimensional implicit schemes (cf. 1171).