Some problems of difference scheme theory

Abstract

1. DIFFERENCE schemes for solving various problems of mathematical physics have appeared in recent years. Economic methods for solving multidimensional problems have been particularly developed (alternating direction schemes, decomposition schemes, locally one-dimensional schemes etc.). A large number of different algorithms is now available for solving a given problem. Special interest is thus aroused by the theoretical and experimental comparison of difference schemes, the development of general principles for constructing families of difference schemes possessing given properties, and finding the schemes in the family that satisfy auxiliary optimality requirements (as regards accuracy, economy etc. 1. There are various schemes which give the same results (are algebraically identical) provided the right-hand sides and boundary conditions are matched in a certain way. In this case our choice of one scheme rather than another must be based on practical convenience. This situation is typical for economic methods of solving multidimensional problems. Several papers 11, 2, 3, 41 have been concerned with comparing economic methods. A comparison in [l, 21 with the operator decomposition method (for the equation of heat conduction in a rectangular region), to which the methods of [5 - 71 are reducible after eliminating the intermediate values, showed that the order of approximation of these methods depends, in the case of non-stationary boundary conditions (and for [71, in the case of stationary conditions also) on the method of specifying the boundary conditions at the intermediate step. It is shown that, by varying the right-hand side at the boundary base-points of the mesh, we can