1549301742-The_Theory_of_Difference_Schemes__Samarskii

600 Econo1nical Difference Schemes for Multidimensional Problems

Thus, the attainable summarized approxi1nation of the additive sche-

me (8) owes a debt to the simultaneous usual approximations and a smn-

marized approximation. In accordance with what has been said above,

equations ( 5) are approxi1nated by the chain of the difference equations

(6)-(7) in a summarized sense and every scheme (8) with the number Cl'

approximates the corresponding equation involved in collection (6) in the

usual sense.

All this enables us to obtain through such procedures a perfect approx-

imation. Let us stress here that the summarized approxin1ation is ensured

by the following conditions:

• the operator L is representable by a sum L = L 1 + L2 + · · · + LP;


• the right-hand side f contains exactly p functions such that f =
  f1 + f2 + · · · + fr ·
  These conditions will be relaxed once we consider
  Jl I'
  L '1l - 2= L('{ 1l = 0( T) ' f - 2= fa = 0( T) ·
  a= 1 a=l

If the operator La includes the derivatives with respect to only one variable
xa, we call it a one-dimensional operator and the equations Pav(a) = 0
refer ,correspondingly, to equations of one variable. The ad di ti ve scheme
(8) is tern1ecl a locally one-dimensional scheme (LOS).
Section 5 of the present chapter will be devoted to such schemes re-
lating to the heat conduction equation.

4. Examples of reduction of inultidimensional problems to chains of one-
   di1nensional ones. It is apparent from that discussion that some class of
   problems for which a solution of problem (6) or problem (11) coincides on
   the grid wh with the exact solution of the multidimensional problem (5)
   plays an important role.

Exan1ple 1 The Cauchy problem

du

-+a.u(t)=O, t>O,

dt

·u(O) -- u 0'

where a. > 0, is good enough for the purposes of the present section. With
the representation a = a 1 + a 2 in view, we may set up the problem