1549301742-The_Theory_of_Difference_Schemes__Samarskii

The quasilinear heat conduction equation^519

with the homogeneous boundary conditions

(30) k+l Vo --^0 '

In this regard, the n1axi1nmn principle states that

(31) II k + u^1 lie< 0.5 I I lf! II (y)/lf! I (Y k )I I e · llu k lie<^2 q llv k lie'^2

k

where q = 05lllf!^11 (y)/lf!^1 (Y)lle < U.5lllf!^11 (Y)llc/c 1 < U.5c 2 /c 1 = q 0 , smce

l(J^1 ( y) > c 1 > 0 and I lf!^11 ( y) I < c 2 •

Using this estimate behind, it is not difficult to establish that for the

convergence of iterations in accordance with a quadratic law, it suffices to

choose the initial approximation so as to satisfy the condition

(32)

The meaning of this is that we should have for the choice y = y = yJ

which is always valid for sufficiently small r.

In practical implen1entations Newton's inethod converges with any

prescribed accuracy E only if

If y > 0, then c 2 = oo and unfortunately the preceding estimates are

meaningless.

However, clue to the maximum principle a solution of the boun<lary-

value problem (29)'-(30) is non-negative:

k+l v = k+l y -y<O, '

provided the condition l(J^11 (y) < 0 holds. The preceding justifies that the

iterations approach the solution from below. Because of this tendency, the

first iteration is such that y < fJ if the initial approximation was taken so

that y < 1]. But it may happen that y < 0, thus terminating subsequent

computations.