1549301742-The_Theory_of_Difference_Schemes__Samarskii

98 Basic Concepts of the Theory of Difference Schemes

2.4 MATHEMATICAL APPARATUS IN THE THEORY

OF DIFFERENCE SCHEMES

1. Some difference fornmlae. In the sequel, when dealing with various dif-
   ference 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 op-
   erators, but there other notations have been used. It performs no difficulty
   to establish a 1·elationship between formulae from Section 2 of Chapter 1
   and those of the present section.

1. Fo7'1nulae for difference differentiating of a product. As
   known from the differential calculus, the formula

holds for differentiating the product of two functions u(x) and v(x).

In Section 1 we have already introduced two types of difference deriva-

tives for grid functions: the left and the right ones, which correspond to

different formulae for difference differentiating of a product

(1)

(2)

(uv) X =u X v+uC+^1 lv X =u X vC+^1 l+uv Xl

( uv ) x -- 1lx v + 1l (-1) vx -- 1lx v (-1) + 1l vx.

Here we retain the notations

j(±l) = f(x ± h),

JC+l) - J

fx = h 1

f - f(-l)

fx = h

In this context, we draw the reader's attention to the fact that these for-
mulae involve the index shift. Let us prove, for instance, the first equality.
With this aim, we write it in the index form
1li+1 Vi+1 - 1t; v,: 1li+1 Vi+1 - 1l; Vi+1
h h
which assures us of the validity of the equality.
2) The s11.1nm.ation by parts form:ulae. We recall the integration
by parts formula
1 1
j uv^1 dx = uvl6 - j u^1 v dx.
0 0