An improved two-layer scheme for sets of parabolic equations with constant coefficients

Abstract

DIFFERENCE schemes of improved accuracy have been proposed for various types of partial differential equations; see [1, 2] for the references. An improved two-layer scheme was described in [2, 3] for sets of parabolic equations with constant coefficients and no mixed derivatives. Here, to obtain a priori estimates, fairly rigid conditions were imposed on the matrix of space operator coef- ficients. It is easy to construct a set of parabolic equations with constant coefficients which does not satisfy these conditions. We show in the present paper, by using Samarskii's general theory of stability of difference schemes [4, 5], that absolute stability, a priori estimates and convergence theorems can be proved for the improved schemes of [2, 3] without supplementary conditions on the space operator. It should be mentioned that the set of algebraic equations obtained may be solved by the matrix pivotal condensation method. In [6], Samarskii's method of regularisation was used to construct absolutely stable three-layer difference schemes of improved accuracy for sets of parabolic equations with constant coefficients and no mixed derìvatives, and their convergence in a uniform metric was proved in the case of a p-dimensional parallelepiped ( p = 2, 3). As distinct from the two-layer scheme of [2, 3], the sets of algebraic equations obtained in [6] may be solved by means of a uniform pivotal condensation algorithm.