1549301742-The_Theory_of_Difference_Schemes__Samarskii

518 Difference Methods for Solving Nonlinear Equations

Under the boundary conditions

(28)

the elimination method is quite applicable in giving kU

1

. There is no doubt
that its stability is ensured by the condition

k

VJ'(y) > 0.

Indeed, this fact is an immediate implication of an alten1ative form of

writing the governing equation

T k+l ( 1 k 2 T) k+l T k+l k

h2 Y i-1 - VJ (yi) + h2 Yi + h2 Y i+1 =Fi

i = 1, 2, ... , N - 1,

k k k k.

where Fi = VJ(Y) - VJ(Y) - VJ^1 (Y )y and y = y^1.

The rate of convergence of iterations can be evaluated by means of

the difference

k+l k+l.

V; = Yi - Yi'

This can be done bv ., inserting yk. z = if,·+ ~- t. 2. and +J.~1^1 = 1/, ~& + k(i.l l in (27)

I k k+l k+l k. I k k ..

VJ (y) V - T u ;ex = VJ(Y) - VJ(Y) +VJ (y) V + T Y.v.v

and taking into account the basic relation

k k k k

VJ(fJ), VJ(Y) + VJ'(y )(y - y) +! VJ11(fJ)(fJ- y )2

and its corollary

k k k k

VJ(fJ) - VJ(Y) + VJ^1 (Y )(Y - fJ) =! VJ^11 (y) v^2 ,

where y = t +et, U < e < 1. The outcome of this is the equation

(29)^1 (k)k+I k+l^1 11 _ k

(^01) k
VJ y v - T u '"'" = 2 .p (y) v - = F , :r = ih, O<i<N,