1549301742-The_Theory_of_Difference_Schemes__Samarskii

366 Difference Schemes with Constant Coefficients

where 1/; = A( O" u + (1 - 2 O") u + O" ii) + <p - utt is the approximation

error of scheme (4) on the solution u = u(x, t) and v = u 0 (x) - '1lt(x, 0)

is the approximation error for the second initial condition Yt = u 0 (x). In

accordance with what has been said above, v = 0( r^2 ). The well-established

expressions u = u + Tut and ii = u - Tut allow us to deduce that

O" U + ( 1 - 2 O") U + O" ll = U + O" T^2 U[t ,

implying that

(7)

Moreover, 1/; = O(h^2 + r^2 ) for an arbitrary constant O", which does not
depend on T and h.
Let ()" = er - h^2 /(12 r^2 ), where the constant er independent of h and
T is so chosen as to provide the stability of scheme ( 4). As can readily be
observed, it suffices to take er>~ (1-c:)-^1 , since scheme (4) is stable for
O" > ~ (1-c:)-^1 - ~ ,-^2 , r = r/h, E > 0. Sche1ne (4) with the 1nember

(8)

h^2

<p=f+ 12.f

II

permits us to make the order of approximation more higher and achieve
1/; = O(h^4 + r^2 ).
The boundary conditions of the third kind

8u(O, t)

0

= {3 1 u(O, t) - μ 1 (t),

x

are approximated by the following difference equations:

where

P1 Ytt = A - ( O" if + ( 1 - 2 O") y + O" y) + <p-,

P2 Ytt = A+ ( O" f; + ( 1 - 2 O") y + O" y ) + <p + ,

A+y =

i -- (^0) '
i -- N '