1549301742-The_Theory_of_Difference_Schemes__Samarskii

316 Difference Schemes with Constant Coefficients

Summarizing, under condition (39) scheme (16) is stable with respect to

the initial data and the right-hand side.

Applying a priori estimate (40) to problem (III) yields

. j ·I
II z^1 +

1

lie < L^7 II1/J^1 lie·

j'=O

We give a brief survey afforded by the above results: scheme (II) converges
uniformly with the same rate as in the grid L 2 (wh)-norm (see (35)) if and
only if condition (39) holds. The stability condition (39) in the space C
for the explicit scheme with (J" = 0, na1nely T < ~ h^2 , coincides with the
stability condition (25) in the space L 2 (wh) that we have established for the
case (J" < ~· The forward difference scheme with (J" = 1 is absolutely stable
in the space C. The symmetric difference scheme with (J" = ~ is stable in
the space C under the constraint T < h^2.

8. The method of energy inequalities. The well-developed method of energy
   inequalities from Chapter 2 seems to offer more advantages in investigating
   the stability of scheme (II) with weights.
   We first illustrate its employment for a differential equation in tackling
   problem (I) with homogeneous boundary conditions

01l o^2 u

8t = ox2+f(x,t),

( 41)

O<x<l, O<t<T,

u(O, t) = u(l, t) = 0, u(x, 0) = u 0 (x).

Here the inner product and associated nonn are defined by

1

(u, v) = J u(x) v(x) dx,

0

llull = J(u, u),

where u(x) and v(x) are functions defined on the segment 0 < x < 1 and
vanishing at the points x = 0 and x = 1. Let us mu! ti ply the equation by
ou/ ot and integrate it with respect to x from 0 to 1. The outcome of this
lS

II

OU 112 ( OU o

2

u) ( OU)

8t + -Ft ' ox^2 = .f' 8t ·

Integrating the second term by parts and taking into account the equality

OU 01l I l - 0

ot ox 0 - '