1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable three-layer sche1nes 445

Comparison of this equation with (79) gives u = l/h^2 > II A 11/4, that

is, the Du Fort-Frankel scheme is stable for any T and h. This is a way

of producing an analog of this scheme for the case when L is any elliptic

operator with u still subject to condition (80).

2) The asymmetric three-layer scheme

3 1 A~

2 Yt - 2 Yr + Y = <p or

3y - 4y + y A~

-----+ y = <p

2r

suits us perfectly for solving the heat conduction equation. Using the for-

mulae

T T T2

Yt = Yf, + 2 Ytt , Yr = Yf, - 2 Ytt , Y = Y + T Yf, + 2 Ytt ,

it is plain to reduce it to the canonical form

( E + T A) Yf, + T

2

(! + ~ A) Ytt + A y = <p '

giving B = E + r A and R = E/r +~A. These assure us of the validity of

the conditions

B>E

for any A> 0. If A= A* > 0, the scheme concerned is stable in the norm

llY(t + r)ll(i) = ~ (lly(t + r)ll~ + lly(t)ll~) + T llYtll~ ·

8. Three-layer schemes with non-self-adjoint operators. The three-layer
   explicit scheme with a self-adjoint operator A

yo+ t Ay = 0, A=A*>O,

is absolutely unstable; the necessary stability condition R > iA is violated

for this case, since R = 0. This scheme is unstable in any nonn II · lln and

refers to a generalization of the well-known Richardson sche111e for the heat

conduction equation

Yi - Yi

2r

Yi-I - 2yi + Yi+l

h2

The implicit scheme Byo + Ay = 0 with any operator B = B* > 0 is also
t
absolutely unstable.