A priori estimates for a family of economic schemes

Abstract

TO solve a multi-dimensional parabolic equation, containing no mixed derivatives, economic difference schemes are described in [1], which are additive and symmetrized and guarantee an accuracy of O( τ 2 + h 2), and in the case of constant coefficients an accuracy of O( τ 2 + h 4), where τ is the network time step, and h the maximum step of the network ω h , in the space variable. This scheme can be treated as a generalization of the method of alternating directions described in [2] to the case of p dimensions. This paper is concerned with obtaining a priori estimates for the solution of the schemes described in [1], including estimates which take account of the special form of approximation error. The estimates are obtained with the usual assumptions about difference operators such as linearity, non-negativeness etc. To estimate the accuracy of concrete schemes the following are necessary: 1) a study of the structure of the approximation error and its evaluation in corresponding norms and 2) a verification that the properties of difference operators are satisfied. In future the evaluations obtained here will be applied to the concrete schemes of [1]. With their help we shall establish the stability and obtain an estimate of the accuracy of the schemes of [1].