1549301742-The_Theory_of_Difference_Schemes__Samarskii

578 Econo1nical Difference Schemes for Multidin1ensional Problems

the following algorithm is recommended for the possible applications:

+1.

yl =y1+rw(2)·

In this context, we should focus the reader's attention on the process of

specifications of boundary conditions by doing Gaussian elimination along

the rows for w(!J and along the colu1nns for w( 2 J° \:Vhen p. happens to be

independent oft, that is, f.-l = p.(x), the quantities w( 11 and w( 2 ) satisfy the

h01nogeneous boundary conditions. · ·

Example 5 In tackling problem (26) the operator L is approximated by

the difference opera.tor

p

Ay= 0.5 L [(kcx,e(x,t)yxJ'),"" + (ka,e(J:,t)yxtl),;;j

a,,6=1

with the coefficients k°' ,6 still subject to the following conditions with con-

stants c 2 > c 1 > 0:

p p p

c1 2= c:J <

"°' - 2=

kcq3(:c, t)~0:~/3 < C2 2..:2 ~u for all .r E (,' 0 , t > 0.

a= l a,,6=1 ex= l

Having stipulated the condition l(ka,e)tl < c 3 , where c 3 = const, the oper·

0

ator A(t)y = -A(t)y is Lipshitz continuous in the space H = r2 h· Also, it

will be sensible to approve the same selection rule for the regula.rizer as in

the preceding example:

p

R = L Ra,

0

Ra Y = -(} C 2 A a Y,

0

AaY= Yx 0 x 0 •

a=l

In that case the factorized scheme (39) is stable under condition ( 40)
or, what amounts to the same, for (} > (1 + c)/4, c > 0, and it is of
second-order accuracy with respect to all the variables.