1549301742-The_Theory_of_Difference_Schemes__Samarskii

The quasilinear heat conduction equation 509

This can be solved by the standard elimination method, whose computa-

tional procedure is stable under the condition f'(y) < 0.

The convergence rate of such iterations needs investigation with regard

to the error

k+l ·u=y-y, k+l

where y is the exact solution of problem (2). Upon substituting two sub-

sequent 1terat10ns.. y k = y + v k an d k+l y = y + k+l ·u. mto equat10n. (6) we may

set up the problem for the error k+l ·u :

( 7) k+l v ,,,. + .I •/ ( Y k ) k+l v = - F k ,

(8)

k k k k

F = f(y) - f(y) + (y - Y) .f'(y) ·

Taking into account the well-established decon1position

k k k k k

f (y) = f (y) + .f' (y) (y - Y) + ~ f" (f;) (Y - y)^2 ,

k k k

where f; = y + 8(y - y ), 0 < {} < 1, we find that

Thus, it is required to evaluate a solution of the problem

(9) k+l V .z:.r + f / (y) k k+l V = 2 1 f II (y ~ )v k^2 ,

If f(y) is a concave function, that is, f"(y) > 0, then due to the

maximu1n principle we inight have

k+l k+l k+l

· v = y -y<O, y <y,

thereby clarifying that the iterations approach the exact solution of problem
(2) from below. It is plain to show that for a solution of problem (9) either
of the following esti1nates

( 10)

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