1549301742-The_Theory_of_Difference_Schemes__Samarskii

576 Econom.ical Diffe1·ence Schernes for Multidhnensional Problerns

by the factorized operator

p

E+2rR= IT (E+2rRa)=E+2rR+4r^2 Qp, R=R+2rQp,

a= l

where Qp = Ri R2 for p = 2 and Qp = Ri R2+R1 R3+R2 R3+2 rR1 R2 R3

for p = 3, etc., the structure of the factorized scheme is

(38) B1 ···BP Yt = -F, Ba = E + 2 T Ra.

The canonical form is given by

(.39)

where B = E + 2 r^2 Qp and R = R + r Qp.

Let now 1/J 0 ( u) be the error of approxi1nation in the class of solutions

1l = 1t(x, t) of the continuous problem for the primary scheme (36) and

1/; 1 (u) be the same quantity for the factorized scheme (39). In this regard,

it should be noted that

"!* y' = ., L.T 2Q p1lt.

When llQP utll( 2 ) = 0(1) is accepted in son1e suitable grid norn1 II · 11( 2 )
built into stability theorems, we might achieve ll1/ill( 2 ) = O(r^2 ) and in
passing from the primary scheme to the economical factorized scheme (38)
the error of approximation changes within a quantity of 0( r^2 ). Following
these procedures, we obtain economical factorized schemes of second-order
accuracy in r as stated before due to the extra smoothness of the solution
u. Such a stability analysis of schemes (36) and (39) is mostly based on the
further treatment of the operators R and A as linear operators acting from
0
the space H = r2 h into the space H. In particular, this means that the
boundary conditions on ih are homogeneous for a scheme approximating
(26).
Under the natural premises

another conclusion can be drawn that the prin1ary schen1e ( 3(3) is stable,
provided the condition

( 40) c > 0)