1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients 313

implying that 1 + O"L\ > E for all k = 1, 2, ... , N - 1 and providing in

combination with (29) the validity of the following estimates:

and

(33) II ~+^1 II

j'=O

For O" = O" * the condition O" * < 0 ensures T < t h^2 and the value E = ~

can easily be found from the equation ~ (1 - E) = / 2. Collecting estimates

(24), (31) and (33) and summarizing the preceding results, we deduce the

following assertion.

If the conditions O" > !--1- h^2 T-l = 0" 0 and O" > 0 hold simultaneously,

then scheme (16) is stable with respect to the initial data and the right-hand

side, so that the solution of problem (16) admits the estimate

When O" < 0, for the stability of scheme (16) with respect to the right-hand

side

1 (1-c)h^2

O">--- 2 4T =O" c>

is a sufficient condition. Here E E (0, 1) is an arbitrary constant indepen-

dent of h and T. For such a choice, the solution of problem (16) satisfies

the estimate

For the sche1ne of accuracy O(h^4 + r^2 ) we thus have E = ~and O"* < 0
if T < t h^2.

6. Convergence and accuracy in the space L 2 (wh). We state here that the
   convergence of scheme (II) follows from its stability and approximation.
   The error z = y - u is just the solution of problem (III). Using a priori
   estimate (31) behind we deduce that

(34)

. j ·I
II z^1 +
1
II < L T II 1/J^1 II as ()" > 0.
j'=O