1549301742-The_Theory_of_Difference_Schemes__Samarskii

490 Homogeneous Difference Schemes for Tirne-Dependent Equations

specifying the difference conditions approximating the boundary conditions

of the third kind. The outcome of this is

Yt = A ( {) ( O" ~I/ + ( l - J) ~I/) + f>, 0 < :r; = ih < l , t = )

0

) T >^0 ,

• 0.5 h Yt,o - aN(t) (J Yx,N + ( 1 - J) Yx,N)

Here P, 1 = μ 1 (t)-0.5hf(O,i) and P, 2 = μ 2 (t)-0.5hf(l,i) fort= ti+0.5r.

The resulting scheme is of accuracy 0( r^2 + h^2 ) for O" = 0.5 and it is of

accuracy O(r + h^2 ) for O" > 0.5.

The best possible choice for later use of the elimination method is due

to

where

u - ~~~~~~~~~~-a^1 ( i)

] - a 1 ( t) + h (3 1 ( t) + h^2 / ( 2 O" r) '

aN({)

(1 - ,O") (-aN(i) Yx,N - f32(f) YN) + 0.5 h YN /r - P2

1

;^2 = O" (aN(t)/h + (3 2 (t) + h/(2 O"r)) '

F = ((1 - O") A(i) y + y/r + <p(t)) 0"-^1.

Having completed the elimination, we observe that the computational pro-
cedure is stable if O" > 0, because 0 < u 1 < 1 and 0 < u 2 < 1.

10. Monotone schen1es for parabolic equations of general form. It is re-
    quired to find a solution of the following proble111 foor a parabolic equation