Heat conduction equation with several spatial variables 349we address the readers to Chapter 6 and Concluding remarks therein. We
note in concluding that scheme (15) is stable for any T and ho: and do not
pursue here analysis of this: the ideas needed to do so have been covered.
The second, no less importance, question is: how to solve the system
of difference equations2
L T ()" 0: Ao: y - y = - F.
o:=lAL first glance, the matrix elimination n1ethod suits us perfectly. But
O(N^2 ) operations are required in its application, where N is the total
number of the nodes of the grid wh. Just for this reason scheme (15) is
unacceptable in practical implementations. We will show later that it may
be replaced by some schemes of the same order O(I h 14 + r^2 ) and O(N)
arithmetic operations are required in this connection for deterrnination of y
by applying successively the scalar elimination for a three-point equation.
The resulting schemes are said to be economical, so there is some reason
to be concerned about this. One needs to exercise good judgment in de-
ciding which to consider. The scheme we have constructed above is aimed
at designing other schemes with the indicated property. Here we will not
elaborate on these matters. If you wish to explore this more deeply, you
might find it helpful to refer to Concluding Rernarks at the very end of the
present chapter and references therein.5.4 SCHRODINGER TIME-DEPENDENT EQUATION- Two-layer scheme with weights. We are now interested in various dif-
ference schemes for the Schrodinger equation
(1)au
z
atll(x, 0) = u 0 (x),
O<x<l, t > 0,
u(O, t) = ll(l, t) = 0,
By analogy with the heat conduction equation we employ the method of
separation of variables, in the framework of which a solution of this problem
is sought as the series
00
u(x, t) = L ck ei>,kt ,.\k(x),
k=l