LOCAL ONE DIMENSIONAL DIFFERENCE SCHEMES ON NON-UNIFORM NETS*

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 ([II- 191). 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 re-spect 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). In this paper we consider local one-dimensional schemes on arbitrary non-uniform nets for linear and quasi-linear parabolic equations with the “heat conduction coefficient ” k, = k,(n, t, u) dependent on the “temperature ” u = a(x, t). The maximum principle, with the help of which the uniform regularity of the problem for the error z = y- u with re-spect to the approximation error was established in [IO], does not hold in the case ka = k,(x, t, u) and in the case of non-uniform nets when k, = k,(x, t) it enables us only to prove first order accuracy with re-spect to h (for problems II, and II,, cf. Section 1). For a number o