1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable three-layer schemes

In this case we agree to consider in (112) er = c 2 /h^2 , Ay

Ay = (ay:r;),:).

For the multiple heat conduction equation

(114)

uJr = 0' u( x, 0) = u 0 ( x) ,

457

-Ay and

G being a. parallelepiped (0 < xc, < / 00 o: = 1,2, ... ,p), 0 < c 1 < ka < c 2 ,

other ideas a.re connected with

p 1

I: fi2 Cl '

where hCI is the step of a. grid wh = { x = (xi' ... 'xp) E c} a.long the

direction x a.

Let Ao= A 1 +A2, where A 1 and A 2 a.re adjoint or "triangular" (with

a. triangular matrix) opera.tors, so that

(Aoy, y) = 2 (A 1 y, y) = 2 (A2y, y).

Setting R = er A 1 or R = er A2 we arrive at scheme ( 108), which is stable for

er> 212 2er 0 (1 2 is a. constant involved in (111)).

Example 2 The a.sym1netric scheme for the heat conduction equation

belongs to the fa.mtly of "triangle" schemes having the fonn

(115)

Here

erT

Yt + h Yxt = A Y ,

Ay = -Ay,

er

Ry= h Yx'

Ay = Yxx ·

Ao =A, 11 = /2 = 1.

Sche1ne (115) is stable for er > l - ~; and conditionally a.pproxi1na.tes the
heat conduction equation to 0( h) as T = O(h^2 ).