1549301742-The_Theory_of_Difference_Schemes__Samarskii

Higher-accuracy sche1nes for Poisson's equation 293

Because of this, the equation

A'y = -<p,

(7)

provides an approximation of order 4 on a solution u = u(x) of Poisson's

equation (1). Indeed, formula (6) gives

A'u + <p = (A'u + <p) - (Lu+ f) = 0(1h1^4 ),

The operator A' is defined on the nine-point "box" pattern (see Fig. 16)

consisting of the nodes (x 1 +1n 1 h 1 , x 2 +m 2 h 2 ); m 1 , m 2 = -1, 0, 1, by means

of which schen1e (7) is representable by

+ -1 ( 1 - + -1 ) (y( +li ,+b) + y( +li ,-12)

12 Ji2 I Ji2 2

where y(±li) = y(xi ±hi ,xJ, y(+li,-lo) = y(xi +hi ,x 2 -h 2 ), etc. On the
square grid (hi = h 2 = h) the final result is sin1ple to follow:

_ 4 (y: + Y2 + Y3 + Y4) + Ys + YG + Y7 + Ys ~ h2

Yo - 20 + 10 'P

(see Fig. 16).
To avoide cumbersome co1nputations, we substitute Ai f for Li f and
A 2 f for L 2 f into the formula for <p having replaced <p by 0(1h1^4 ), which
does not change the order of approxi1nation for 1/; = A'u + <p = O(I h 14 ), so
that