1549301742-The_Theory_of_Difference_Schemes__Samarskii

480 Homogeneous Difference Schemes for Time-Dependent Equations

where lliJJI = J(l,17^2 ] and llzll < llvll + llwll, we get for CJ> CJ 0 , CJ£

0.5 - (1 - c:)/(Tll A II), the estimate

(55) 11<^1 " lie <

2 ~, { llry

0

J I + llry^1 ] I +

1

t, r ll'lf' 11}

In what follows it is supposed that k(x), f(x, t), u 0 (x), u 1 (t) and u 2 (t)

are smooth functions and the conditions under which

(56) 1) = O(/hl^2 +Tm~), 171 = O(lhl^2 +Tm~), 1/J* = O(lhl^2 + T^2 )

hold. Then schen1e ( 49) converges unif01·n1ly on any sequence of nonuniform

grids {w 11 } with the rate O(rm~ + lhl^2 ), where lhl = nlax hi, if CJ> CJ<.

l5,i5,N - '

This fact follows immediately from the combination of estimates (55) and

(56).

We touch upon briefly the convergence of scheme (49) in the class of

discontinuous coefficients and will pursue some analogy with the stationary

case which has been considered on the same footing in Chapter 3, Section

4 under the following assumptions:

(a) the functions k(x) and J(:r, t) n1ay have only a finite nmnber of

discontinuities of the first kind on straight lines parallel to the co-

ordinate axis Ot;

(b) the grid w1i = wh ( K) is chosen in such a way that all of the discon-

tinuity lines of the functions k(x) and f(x, t) will pass through the

nodal points;

(c) the functions k(x), f(x, t) and u(~'., t) have all necessary derivatives

in the regions lying between the discontinuity lines so that for1nulas

(53)-(54) and estimate (56) are still valid at all the nodes of the

grid w~ 1 (K).

We note in passing that 'Pi is determined by formula (47), due to

which scheme (49) converges uniformly with the rate O(Tm~ + lhl^2 ) under

the aforementioned conditions on sequences of special grids.

Remark 1 Convergence in the norn1 of the grid space L 2 occurs with the
same rate if condition c) is relaxed:

(57)