1549301742-The_Theory_of_Difference_Schemes__Samarskii

Economical factorized sche1nes^565

Knowing the value y = y1 on the jth layer, it is required to find the value

y1 +^1. In preparation for this, we derive the equation related to y1 +^1 with

a known right-hand side Fj:

As can readily be observed, 0( N) operations are needed iu giving F.i and

their amount is proportional to the total nurnber of the grid nodes. This

is certainly so with any difference scheme, whose pattern is independent

of the grid. From equation (2) it is easily seen that the stable 6Che111e (2)

will be "economical" once the users perform O(N) operations while solving

equation (2).

Let "econon1ical" operators B 0 , ct = 1, 2, ... , p, be such that 0( N)

operations are necessary in connection with solving the equation

(3) Bex v = F.

Then scheme (1) with a factorized operator B of the structure

(4)

will also be "econo111ical" in line with e6tablished priorities: the nmnen-

cal solution to equation (2) with operator ( 4) requires O(N) operations.

Indeed, a solution to the equation

(5)

can be found by successive solution of p equations of the form (3):

(6) B1 Y(l) = F^1 , Bu !f(o) = !f(u-1), CY= 2, :3, ... ,Ji,

• ··o^0 that yj+l -- y (p) · He1·e y ( 1 J -- yj+l/p ' · · · ' y (a J -- yi+cx/p ' · · · ' y (p -1) --
  yj+(p-l)/p stand for intermediate values arising in the process of calcula-
  tions. It follows from the foregoing that the stable scheme ( 1) with the
  factorized operator B being a product of a finite number of "economical"
  operators B 1 , ... , Bp becomes "economical", so there is some reason to be
  concerned about this. All the sche111es with a factorized operator B are
  called factorized schemes. It was shown in Section 1 that the econo111i-
  cal alternating direction scheme is equivalent to the factorized sche111e with
  the operator

(7) B=B1B2, Bcx=E-0.5rAcx, AaY=Yx,,,x,,,> CY=l,2.