1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with several spatial variables 345

2. The explicit three-layer scheme. We now turn to the simplest explicit
   three-layer scheme known as the Richardson scheme and being an analog
   of scheme (59) from Section 1:

(12)

However, it is absolutely unstable. Substitution of the half-slllTI

for y{, where

Ytt =

into the right-hand side yields the p-climensional analog of the Du-Fort-

Frankel scheme. The forthcoming substitution

p

• 2 """"' ~ h2^1 y1(x).

leads to an alternative form of writing

p 1

A y = T2 """"' ~ h2 Ytt ,

a=I a

which will be involved further in the explicit sche1ne

(13)

T2 6_

yo+ t 4 Ytt = A Y + r.p ,

where

p 1

6. = (^4) 2= h2
   CY= 1 CY
   In the case of a cube grid with h 1 = h 2 =
   designing one more sche1ne instead of (13):
   ( 13')
   p T2
   Y^0 t + - 1 2 -? Ytt = A Y + r.p
   CY=l CY
   h one succeeds in
   and in establishing as its immediate irnplication the useful formula for de-
   termining iJ = yi+^1 :
   (1 + 2 I) iJ = (1 - 21) y + 4 I y + 2 TA y + 2 T r.p,