1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients 325

Keeping T = ah (1 + O(h)) with CL = const we draw the conclusion that

scheme (62) approximates any equation of the form

OU 82 u 82 u

+ CL2 --

ot 8t^2 8x^2

The following implicit three-layer schemes with weights are quite applicable

in solving equation (3):

• symmetric schemes

(63)

• nonsymmetric schemes

(64) Yt + (} T Ytt = A y + zp.

Equations (63) and (64) contain the three layers (tj_ 1 , t 1 , tj+ 1 ), so that, in

what follows, tj > r, j > 1. The value y(x, 0) = tt 0 (x) is known in advance

and the value y(x, r) should be preassigned in addition to the available

information. For example, this value can be determined either by

or y(x, r) = y(x, 0) + ru 0 (x)

with u 0 (x) = u~(x)+ f(x, 0) still subject to the condition y(x, r)-u(x, r) =

0( r^2 ) (see Chapter 2, Section 1).

Sometimes two-layer schemes are effectively used to assign the value

y(x, r).

By virtue of the asymptotic relation

(} U + ( 1 - 2 (}) tl + (} U = tl + (} T^2 tl[t = tl + 0( r^2 )
the symmetric scheme (63) is of order 2 in T and h for any (}. The error of
approximation for scheme (64) is representable by

which serves to motivate the accuracies which interest us in the following
cases:

1/J = O(h^2 + r^2 )

1/J = O(h^2 + r)

for

for

zp= f,

zp = J.