Homogeneous difference schemes

Abstract

Homogeneous difference schemes, suitable for transforming differential equations whose coefficients belong to certain classes of function into difference equations, are defined and discussed. The main points which arise are, first, whether the solution of the resulting difference equation converges to that of the original differential equation in the given class of coefficients, and of what order the convergence is, if it exists; and, secondly, how the “best” scheme, giving a high degree of accuracy in the widest class of coefficients and stability with respect to computing errors, can be selected. A basic lemma concerning the necessary condition for convergence is proved. Examples are given of a difference scheme for Sturm-Liouville type operators in the class of sufficiently smooth coefficients, of a scheme for the first boundary problem in the class of smooth coefficients and in the class of discontinuous coefficients, as well as in the class of piece-wise continuous coefficients. The latter is the basic class of coefficients which is discussed in the article. Green's function for the difference operator is constructed, and bounds are found for it and for its first difference ratios.