Homogeneous difference schemes of a high degree of accuracy on non-uniform nets

Abstract

The homogeneous difference schemes corresponding to the boundary problems for the differential equation l (p,q,f) u = d dx [ 1 p(x) du dx ] − q (x) u+tf (x) = 0, 0 < x < 1 . (1) M 2 ⩾ p ( x) ⩾ m 1 s> 0, 0 ⩽ q ( x) ⩽ M 3, with piece-wise continuous coefficients ( p, q, f gE Q (0)) include an exact scheme [1], [2]. This scheme enables us to determine the net function which coincides with the exact solution of the boundary problem on an arbitrary non-uniform net s n ( x 0 = 0, x 1, …, x i , …, x N = 1, h i = x i − x i i−1 ). In this article we construct schemes of any (pre-set) order of accuracy on non-homogeneous nets. All the schemes will be of the form l h (p,q,f)y i = 1 kh i δ( ▽y i h ia i h) − D i hy i, + gF i h , kh i = 0,5 ( h i + h i+1 ), Δy i = ▽ y i+1 = y i+1 − y i . (2) The upper index h is the conventional notation for the dependence between the coefficients and the net. In the case of a uniform net ( h i = h = 1 N , i = 1, 2, …, N ) the difference schemes (2) belong to the family of homogeneous three-point conservative schemes [2], [3]. This is also true when the concepts of homogeneity and conservativeness of schemes are generalized to non-uniform nets. Schemes of an increased degree of accuracy prove to be especially useful in a number of cases which are important in practice, such as in the solution of equation (1) with piece-wise constant coefficients with a large number of discontinuities, or in the solution of systems of such equations, or in the solution of equations of heat conduction and diffusion of the form δu dt = L (p, q, f)u .