Homogeneous difference schernes for hyperbolic equations 4997.2 HOMOGENEOUS DIFFERENCE SCHEMES
FOR HYPERBOLIC EQUATIONS- The original problem. In a co111mon setting it is required to find in the
rectangle
Dr= [O < x < 1] x [O < t < T]
a solution of the first boundary-value problen1 for a second-order equation
of hyperbolic type
(1)(^02) u. )
ot 2 =Lu+f(x,t, Lit= ox o ( k(x,t)ou)
0 x,
(x, t) E Dr,
(2) ll ( X , 0) = U 0 ( X) ,
(3)
() < C 1 < k ( X 1 t) < c 2 1
where Dr = ( 0 < x < 1) x ( 0 < t < T], under the following assumptions:
the problem is uniquely solvable, its solution is continuous in the closed
doma.in Dr and possesses all necessary derivatives which do arise in the
further develop111ent, the coefficient k(i:,t) and the right-hand f(x,t) ma.y
have discontinuities of the first kind on a finite number of straight lines
parallel to the axis Ot ( "im111ovable discontinuities"), on every discontinuity
line x = ~s, s = 1, 2, ... , s 0 , the conditions of conjugation relating to the
continuity of the functions tl and k ou/ ox for x = ~s, s = 1, 2, ... , s 0 , must
hold:
(4) [tt] = tl((, + 0, t) - tl(~s - 0, t) = 0, [kou/oJ:] = 0.
- Homogeneous difference sche1nes. In preparation for designing a h01no-
geneous weighted sche111e associated with proble1n (1)-(3), let
be a nonequidistant grid on the segment 0 ~ ;r < 1, wT = {tj = ir,
j = 0,1,2,. .. ,j 0 } be an equidistant grid on the segment 0 < t < T and
let a suitable grid in the rectangle Dr be 111ade up by whT = wh x WT' A
homogeneous difference scheme for solving problem ( 1 )-(3) can be obtained