1549301742-The_Theory_of_Difference_Schemes__Samarskii

328 Difference Schemes with Constant Coefficients

the minimal expenses of execution time. A number of operations may be

minimized by improvements of difference schemes as a result of modeling

the basic properties of a differential equation in the space of grid functions

as well as possible. One of such typical properties is the true asymptotic

behavior of its solution as t-+ oo.

To illustrate our approach, let us consider, for example, the heat con-

duction equation

OU

at

(1) O<x<l, t > 0,

u(x, 0) = u 0 (x), u(O,t) = u(l,t) = 0, t > 0.

The problem concerned can be solved by the method of separation of vari-

ables, within the framework of which we seek its solution as a sum

where

00

u(x, t) = L ck e->.kt Xk(x),

k=l

1

ck= ( Uo, xk) = J uo(x) Xk(x) dx,

0

co 00

11 u ( x, t) 112 = L c~ e - 2 >. k t 11 x k 112 = L c~ e - 2 >. k t,

k=l k=l

since 11 X k 11 = 1. As )..k increases along with the subscript k, we thus have
)..k > \ and
co
II u(t) 112 < e-2>.1t L c~ = e-2>.1t II Ua 112,
k=l

which means that the solution of problem (1) admits the estimate

(2) llu(t)ll < e->.it llu(O)ll for any t > 0.

With increasing t the harmonics u(k) =ck e->.kt Xk(x), k > 1, damp
more rapidly than the first one, so that for sufficiently large t

(3)