1549301742-The_Theory_of_Difference_Schemes__Samarskii

386 Stability Theory of Difference Schemes

equation (3) can be recast as

(4) Byt+Ay=VJ(t), y(O) = Yo E Bh.

We call both equations (3) and ( 4) the canonical form of two-layer

schemes. Equation ( 4) is sirnilar to the differential equation

cl u

B dt + Au = f(t)

Example 1 For the heat conduction equation

au

7ft = Lu+f,

we made in Chapter 5 the design of the two-layer scheme with weights

• Yt = A ( O" y + ( 1 - O") y) + 'P , Av= (a(x)vx):r:.
  With the aid of the the identity

we reduce it to

A y-y

y=y+r--

T y + TYt

Yt - O" T A Yt - A y = 'P.

Comparison of this equation with equation ( 4) gives

B=E+O"rA, A= -A.

This form of writing reveals what is meant by the canonical form of a
weighted two-layer scheme.
At the next stage we proceed to solve equation ( 4) with respect to
y = Yn+l · If the inverse operator B-^1 exists, one can write down

(5) y = Sy+r<f;, S = E-rB-^1 A,

The operator S is called the transition operator (from one layer to an-
other). In addition to the canonical form ( 4), alternative forms of writ-
ing will appear in the sequel for two-layer schemes: By = Cy+ T 'P or
B Yn+l = C Yn + T 'Pn, where C = E - TA, E being the identity operator.
In the case B = E, scheme (3) is called an explicit two-layer scheme:

Yn+l - Yn +A

T Yn = 'Pn'

permitting us to determine the value Yn+l on the upper layer by the formula
Yn+l = Yn - T Ay 11 + T 'Pn· If B # E, then scheme (3) is called an implicit
two-layer sche1ne.