The alternative-triangular method 691k+l
giving the values f; and lJ at the centre (£ 1 h 1 , i 2 h 2 ) of the suitably chosen
pattern.
Further development with '.Von the grid wh is due to a special choice of
the near-boundary node i 1 = 1, i 2 = 1 at the left lower corner of the domain
of interest so that two other nodes ( (i 1 - 1) h 1 , i 2 h 2 ) and (i 1 h 1 , ( ( i 2 - 1) h 2 )
should belong to the boundary on which the values f;; 1 _ 1 and f;; 2 _ 1 are
already known. Formula ( 42) gives the value f; with further contingency
either along rows or along columns.
The directions of row account are fro111 the left to the right: for fixed
i 2 = 1 we are moving from i 1 = 2, 3, ... to i 1 = N1 - 1 and, after this,
for fixed i 2 = 2 - from i 1 = 1, 2 ... to i 1 = N1 - 1. The directions of
k+l k+l
column account are bottom up. The value lJ depends on lJ ii+l and
k+l lJ i h. h
2 +^1 , therefore account starts at t e ng t upper corner of the domain:
i 1 = N1 - l and i 1 = N 1 - 1, so that two adjacent nodes should belong to
k+l A+l
the boundary on which the values lJ ii+l and lJ i 2 +i are already known.
Further calculations are performed either along rows (from right to left) or
along columns (fron1 top to bottom).
All of the calculations are best conducted by recurrence fornrnlas,
whose algorithn1 is certainly stable and is called "through execution"
algorithm.
Within its fran1ework it is necessary to perform 4 operations of addi-
tion and 6 operations of multiplication at every node of the grid for detern1i-
nation. o f k+l lJ wit. h I (now 1 l ec ge o f' F. " I n g1vmg.. F "^10 operations. o f a. d c l' rtron.
and 10 operations of multiplication should be performed. Summarazing,
it is required to carry out 14 operations of addition and 16 operations of
mu 1 tip , l' 1catron. rn. passmg. from lJ k to k+I lJ.
In trying to make these numbers small enough, it is highly recom-
mended to save two numerical sequences instead of t. This can be clone
using the algorithm( 44) wl /h = o )
k
w I -y h = o,k k
where 1.: = Av + f and v I -Yh = μ, making it possible to perfonn 10 op-
erations of addition and 10 operations of multiplication at one node for