1549301742-The_Theory_of_Difference_Schemes__Samarskii

638 Economical Difference Schemes for Multidimensional Problems

The initial condition is satisfied exactly:

(88) y ( x , 0) = u 0 ( x).

The following system of equations emerges in detennining yl +a/(^2 P) = y (a)

and yi+ai/(^2 P) = Y(ai) during the course of the elimination:

tions

Ci-l

F~ = T ~ A~f3Y(p) +r<.p~ +Yca-1)'

f3= 1

p

y+ Ci 1 = T ~ L._.

Having completed the elimination, we 1nust solve the systen1 of equa-

In light of the special structure of the diagonal matrices k~a' whose blocks

are lower triangle, the components Y(n)' 8 = 1, 2, ... , n, of the unknown

vector y 10 ) are to be detennined successively by the elimination method in

passing fro111 c~ to 0: + 1 and from s to s + 1. By the eli1nination fonnulas for

a three-point equation we constitute in a tern1-by-tenn fashion the vectors

Y(a)i 0: = 1, 2, ... ,p. l'vioving in reverse order fro111 ct+ l to 0: and fro1n

s + 1 to 8 the vectors Y(p+l)' ... ,y( 2 P) are recovered from the system

Upon tenninating this process the resultant vector y ( 2 P) is just the solution
y1+^1 = y( 2 P) on the layer t = tj+l·
Since the system of differential equations ( 56) approximates equation
(50) in a summarized sense in compliance with approximating equation
(85) by equation (86) with the number 0:, the additive scheme (86)-(88)
generates an approximation of 0( T + lhl^2 ):

p

·t/J = ~(\ii~ + 4Jt) = 0( T + lhl^2 ).

a= J