The quasilinear heat conduction equation 517- A difference sche1ne. Newton's Inethod. vVe now proceed to constructing
difference schemes for the quasilinear heat conduction equation.
For this, it seems unreasonable to employ explicit schemes with fastly
varying ingredients k ( u), c( u) and f ( u). The power functions of tempera-
ture reflect in full measure the difficulties involved in such a case. For any
implicit scheme one possible stability condition
T 1 min c(u)
-<-h2 - 2 max k(u)is connected with successive step refinement in t to a considerable extent.
Quite often, it depends on the values of k and c in a smaller number of nodal
points. This supports the conclusion that explicit schemes are useless for
our purposes. In an attempt to fill that gap, a considerable amount of effort
has been expended in designing unconditionally stable implicit schemes.
In order to understand some things a little better, the governing equa-
tion is put together with the boundary conditions(25)
iJVJ(tl)
8tu ( x, 0) = 1l 0 ( x) , u ( U, t) = μ 1 ( t) , u (l, t) = μ 2 ( t).
A nonlinear difference schen1e with respect to yj+l(26)X = X; = ih, O<i<N, hN = 1,
may be employed in such a setting under the conditions VJ^1 (y) > c 1 > 0 and
!VJ^11 (y) I < c 2 , providing its stability and convergence in the space C with
the rate 0( T + h^2 ) .• The proof of these assertions is somewhat lengthy and
cun1bers01ne and so it is omitted here.
The nonlinear equation
is aimed at detern1ining yj+l on every new layer by n1aking several iterations
of Newton's method
(27) VJ(Y) k + 1/)^1 (Y) .k (k+l Y - Y k) - T J/xx ·+1 = VJ(y1). ·