1549301742-The_Theory_of_Difference_Schemes__Samarskii

464 Homogeneous Difference Schemes for Time-Dependent Equations

The following implicit schemes are frequently encountered in the the-

ory and practice:

a) the symmetric scheme (o- = 0.5)

Yt = 0.5 A (y + y) + <p;

b) the forward scheme or purely implicit schen1e ( o- = 1)

Yi= Ay + <p ·

3. Stability and convergence. The general stability theory for two-layer
   schemes applies equally well to the stability analysis of the weighted scheme
   (7). With this aim, the appropriate difference scheme with the homoge-
   neous boundary conditions comes first:

(9) Yt = A ( o-y + ( 1 - o-) y) + <p , x E w h , t > 0 ,

y(x,O)=u 0 (x), y=O for x=O,x=l.

All the tricks and turns remain unchanged: we, first, introduce the

0

space SL of all grid functions given on the grid wh and vanishing on the

0

boundary for x = 0 and .i: = l and then define in that space H = SL an

inner product

N-l

(y,v) = L: Y;V;h

i= 1

and associated norm II y II = v1.Y:Y). By means of a linear operator A

acting in accordance with the rule

Ay=-Ay=-(ay 5 ;)," for yEH

the preceding scheme (9) can be rewritten as

( 10) Yt + A ( o-y + ( 1 - o-) y) = <p , t = j T > 0 , y( 0) = 'll 0.

Observe the operator A so defined is self-adjoint and positive definite:

where
fJ = min AdA),
I<k<N-l