1549301742-The_Theory_of_Difference_Schemes__Samarskii

492 Hornogeneous Difference Schernes for Time-Dependent Equations

To equation (79) there corresponds the purely i1nplicit (four-point)

homogeneous scheme

(80) p( x Ji) Yt = A( i) y + \0 ,

y(x, 0) = u 0 (x), y(O, t) = tt 1 (t), y(l, t) = u 2 (t),

where the coefficients p and <pare calculated by the same formulas as before

for d and b±. A special structure of the operator A built into this scheme

results in the error of approxi1nation 1/J = 0( T + h^2 ) and may be of help in

achieving this aim.

The maximum principle applies equally well to the estimation of the

problem (80) solution with zero boundary conditions y 0 = YN = 0 in tack-

ling the governing eLJ.Uation in the canonical form

(Pi/T + CYi + f3i + d;) Yi= CYi f!i-1 + f3i iJ.z+1 +Fi,

Fi = Pi Yd T + \0 i , CY·= z a.(u z z - hb-:-)/hz^2 J

The conditions of Theore1n 3 in Chapter 4, Section 2 are easily verified for

this equation, due to which we might have

Substituting here the appropriate expressions for Fi and D; yields

which assures us of the validity of the estimate

j

11~v+

1

llc < llv°llc +cl L T ll<F

1

llc.

l j'=O

This provides enough reason to conclude that scheme (80) converges uni-
formly with the rate 0( r + h^2 ).