1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternating direction n1ethod 553

Theore111 1 Sche1ne (22) is stable with respect to the initial data and

the right-hancl side. A solution of problem (22) satisfies a priori estimates

(26)-(27).

4. Stability and accuracy. By utilizing the fact that scheme (9)-(14) is
   equivalent to scheme (20)-(21) subsequent considerations of convergence
   and accuracy of the first scheme will appear for the second one. Let 1l =
   u(x, t) be a solution of problem (7) and y = y(xi, tn) be a solution of
   problen1 (9)-(14) and scheme (20)-(21). Upon substituting y = z + 1l into
   (20) we tnay set up the proble1n for the error of this sche1ne:

(29) B zt = A z + 1/J, x E W;, , 0 < t = nr < t 0 ,

zl1'h =0, z(x,0)=0,

where B = ( E - 0.5 T Ai) (E - 0 .. 5 T J\ 2 ) and ijJ is the error of approximation

equal to

From such reasoning it see1ns clear that

lhl^2 = h^2 I + h;' -

under the condition that the solution 11 = v.(:r, t) possesses in the region

Qr = Go x [O, t 0 ] the derivatives

(31)

84x

ox^4 < M.

2

The final result is an in1111ecliate in1plication of the asymptotic relations
0.5 (u + u) = u + O(r^2 ) and ut = u + 0( r^2 ), where u = u(x, tn + 0.5 r)
and the quantity A 1 A 2 1L is bounded. Since for problen1 (29) estimate ( 26) is
valid for the assigned value z(O) = z 0 = 0, we 111ight formulate the following
assertion.

Theorem 2 Under conditions (31) scheme (20)-(21) converges in the grid
norm (23) with the rate O(r^2 + lhl^2 ).