1549301742-The_Theory_of_Difference_Schemes__Samarskii

554 Economical Difference Schen1es for Multidimensional Problems

5. A sche1ne for the governing equation with variable coefficients. The
   intention is to use the alternating direction sche1ne for the heat conduction
   equation with variable coefficients

(32) (x, t) E Qy,

ulr = μ(:i:, t)'

In such setting, for any t, the operator Le, is approximated by the difference

operator

where ao: can be founded either by the formula ao: = k~-^0 ·^50 ) or by the

formula ac, = 0.5 (ko: + k~-^10 >), o: = 1, 2, which guarantee a second-order
approximation provided by the operator Ac,:

Instead of (9)-(10) we are trying to adopt another sche1ne

(33)

n+l -

Y - Y A (t-) - A (t ) n+l^71

0 .0 c T = l Y +^2 n+l Y + 'P ,

t = in+l/2' y(x,O) = u 0 (x),

with the boundary conditions

yn+l = 11 n+l for 'l2 -- U, 'l2 -- Ih
'
(34)

y= μ for 'l 1 - -^0 ' 'l (^1) = N1 ,