1549301742-The_Theory_of_Difference_Schemes__Samarskii

Iterative alternating direction rnethocls 711

The numerical solution of problem (1) by means of iteration schemes can

be done using the alternating direction scheme for the heat conduction

equation au/at= t3.u + f(x), taking now for j = 0, 1, ... the form

(2)

ui+1/2 _ yi

(1)

Tj+l

j+l j+l/2

y ~ 2 ~ =Ai yi+^1 /^2 + A2 yi+^1 + f(x),

Tj+l

with any initial data y^0 = y(i:, 0). As usual, j is the number of the iteration,

yi+I/^2 is the intermediate iteration, r}~l 1 > 0 and r}; 1 > 0 are the iteration

parameters which will be so chosen as to provide a minimal nmnber of

iterations.

The transition fro111 the ith iteration to the (j + 1 )th iteration can be

done by the elimination n1ethod along the rows as well as along the colmnns

for the following three-point equations:

,j+l/2 y - T(l) j+l A l ,j+l/2 y -- Fj '

(along the rows)

Fj = yi + T(l) J +l A 2 yj + T(l) J +1 J '

(along the columns)

Fj+l/2 = ,j+l/2 y + T(2) J+l A 1 y ,j+l/2 + T(2) J+l 1·.

Thus, the users must perfon11 O(l/(h 1 h 2 )) arith111etic operations in calcu-
lating one iteration or 0 ( l) arithmetic operations at every node of the grid
wh.
By analogy with the nonstationary heat conduction equation the it-
erative process (2) is referred to as the alternating direction method
(ADM), the convergence of which can be established on account of the