Mathematical apparatus in the theory of difference schemes

Some difference formulae. In the sequel, when dealing with various difference expressions, we shall need the formulae for difference differentiating of a product, for summation by parts and difference Green s formulae. In this section we derive these formulae within the framework similar to the appropriate apparatus of the differential calculus. Similar expressions were obtained in Section 2 of Chapter 1 in studying second-order difference operators, but there other notations have been used. It performs no difficulty to establish a relationship between formulae from Section 2 of Chapter 1 and those of the present section. 

In Section 1 we have already introduced two types of difference derivatives for grid functions the left and the right ones, which correspond to different formulae for difference differentiating of a product 

In this context, we draw the reader s attention to the fact that these formulae involve the index shift. Let us prove, for instance, the first equality. With this aim, we write it in the index form 

2) The summation by parts formulae. We recall the integration by parts formula 

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1) Formulae for difference differentiating of a product. As known from the differential calculus, the formula 

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Most of the propositions in Chapters 2-4 are of independent value although, for the present book they are used only as part of the auxiliary mathematical apparatus. Some of them were known earlier and the rest were discovered and proven in recent years in connection with the rapid development of the theory of difference schemes. 

One of the popular branches of modern mathematics is the theory of difference schemes for the numerical solution of the differential equations of mathematical physics. Difference schemes are also widely used in the general theory of differential equations as an apparatus available for proving existence theorems and investigating the differential properties of solutions. 

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