1549301742-The_Theory_of_Difference_Schemes__Samarskii

306 Difference Schemes with Constant Coefficients

4. Stability with respect to the initial data. Let us investigate the stability
   of scheme (II) with homogeneous boundary conditions by the method of
   separation of variables. For this, we proceed as usual. This amounts to ap-
   plying to scheme (II) with homogeneous boundary conditions the identities

y = y + T Yt,

and reducing them to the following ones:

(16)

Yt - (} T A Yt = A y + 'P ,

y(O, t) = y(l, t) = 0,

y( X, 0) = U 0 ( X) ,

(x,t)Ewhr>

Scheme (16) is said to be stable if for a solution of problem (16) the

estimate holds:

(17)

where M 1 , and M2 are positive constants independent of hand r, II ll(l)

and 11 · II ( 2 ) are suitable nonns on the grid w h.

With <p = 0 incorporated, the estimate

(l 8)

expresses the stability of scheme (16) with respect to the initial data. When

y(x, 0) = 0, the meaning of the stability of scheme (16) with respect to the

right-hand side is that we should have

( 19) II y(t) 11(1) < M2 max 11'P(t')11(2).

O~t'<t

The stability of scheme (16) with respect to the initial data and the right-
hand side is ensured by estimate (17), valid for the solution of problem
( 16).
Before giving further motivations, let us represent the solution of prob-
lem (16) as a sum y = fJ + y, where fJ is a solution of the homogeneous
equation

(16a) Yt-(}rAyt=Ay, y( 0, i) = y( l, i) = 0 , y( X, 0) = U 0 ( X) ,

and fl is a solution to the nonhomogeneous equation with the initial condi-

tion f!(x, 0) = 0:

( 16 b) Yt - (} r A Yt = A y + <p , y(O, t) = y(l, t) = 0, y(x, 0) = 0.