1549301742-The_Theory_of_Difference_Schemes__Samarskii

74 Basic Concepts of the Theory of Difference Schemes

The meaning of Lh v = vxx as the difference approxin1ation of the operator

L v = v" reveals that

h2 h4

L1 v = v" + - u<^4 l + -v<^6 l(x + Gh)

' 12 :360 '

-1 < e < 1.

We substitute here v = u, u" = - J, uC^4 ) = -f" and accept J" = 0,

implying that J(k) 0 for all k > 2. Then Lh u = u" = Lu, that is,

the approximation error happens to be from the class of solutions to the

equation Lu = - f, where f is a linear function identically equal to zero:

·tf; _ 0. In that case the approximmation is said to be exact. But if J" op 0,

the accuracy will be improvable once we atten1pt the operator in the form

~ h2

L h v = L h v +

12

f 11 ,

thereby providing the approxin1ation ~ = Lh u - Lu= O(h^4 ).
Thus, the accurate account of the error of the difference approxi111ation
on a solution of the differential equation helps raise the order of approxi-
1nation.

4. The statement of a difference problem. In the preceding sections we were
   interested in approximate substitutions of difference operators for differen-
   tial ones. However, n1any problen1s of 111athematical physics involve not
   only differential equations, but also the supplementary conditions (bound-
   ary and initial) which guide a proper choice of a unique solution fron1 the
   collection of possible solutions.
   The complete posing of a difference problem necessitates specifying the
   difference analogs of those conditions in addition to the approximation of
   the governing differential equation. The set of difference equations approx-
   imating the differential equation in hand and the supplementary boundary
   and initial conditions constitute what is called a difference scheme. In
   order to clarify the essence of the n1atter, we give below several exan1ples.

Example 1. The Cauchy problem for an ordinary differential equa-

tion:

(32) u'=f(x), x > 0' u(O) = 1t 0.

We proceed, as usual, on the simplest uniform grid wh
1, 2, ... } and put the difference problem

{

Yx = 'P,

Yo = Uo,

Yi+1 - Yi

or h ='Pi' i=O,l,. ..

ih, z

Yo = Uo,