1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients 323

approximation happens to be the same as we obtained for the boundary-

val ue problen1 of the third kind.

Let us reduce condition (51) to an alternative form convenient for

current manipulations. The outco1ne of solving (51) with respect to y 0 =

Yd +^1 is

(57)

(}

b.. I ,

Along these lines, condition (55) becomes

(}

b.. ,

2

h2

b..l = (J(l + f31 h) + - ,

2r

(58)

// 2 = b,.1 { h2 }

2

( 1 - (}) Y N - I + [ 2 T - (1 - (}) ( 1 + f32 h) l Y N + h il2 ,

so that 0 < xcx < 1 for f3cx > 0, (} > 0 and CY= 1, 2. In the determination
of y on a new layer we obtain the difference equation (8) with boundary
conditions (57)-(58). This problem can be solved by the right elimination
method (see Chapter 1, Section 2. 5).
Stability of scheme (II) with the third kind boundary conditions can be
discovered following established practice either by the method of separation
of variables or on account of the tnaximum principle.

10. Three-layer schemes for the heat conduction equation. One of the
    first schemes arising in numerical solution of the heat conduction equation
    ou/ ot = 82 u/ 8x^2 was the Richardson explicit three-layer scheme such as

(59)

where

y-y

tjo • t = 2 T

or

. if= iJj+I. ,

yo t = Ay,

y = yJ, Ay = Yxx ·

By exactly the sa1ne reasoning as before, it is not difficult to verify that this
scheme is of order 2 with respect to T and h both: 1/J = A u-uo = O(h^2 +r^2 ).
t