Homogeneous difference schemes for hyperbolic equations 501The second one is connected with the difference equation for determi-
nation of y( T):(9^1 ) (E-O"r^2 A(O))Yt(x,O)=u 0 (a:)+0.5r(Au 0 +f(x,O)).
Summarizing, the homogeneous difference scheme (7)-(g) or (7), ( 8),
(9^1 ) is put in correspondence with the original problem (1)-(3) under con-
sideration.
The computational procedures for three-layer schemes were estab-
lished before. For their successful realization we need to know the values yj
and yj-l on two preceding layers for searching f} = y1+^1 by the elimination
method being used on every new layer t = tj +i in solving the boundary-
value problen1 with respect to f} = yj+l:(10) (E-O"r^2 A)f}=F, O<x=ih<l, f} 0 =u 1 , YN=i12,
F(t)=2y-y-r^2 A((20"-l)y-O"y)+r^2 <p, t>r,
F(O) = u 0 + r^2 (0.5 - O") A(O) u 0 + ru 0 (x) + 0.5 r^2 f(x, 0).
- The error of approximation. Let u(x, t) be a solution of the original
problem (1)-(3) and y(x;, tj) =if; be a solution of the difference problem
(7)-(9). The next step is to set up the difference proble1n for the error
zf =if; -1li, where u{ = u(x;, tj), by inserting in (7)-(9) the sum y = z+1l:
( 11) (E-O"rA)zr^2 1 =A:'+4,(:c,t), O<:c<l. t>O,
z(x, 0) = U, z(O, t) = z(l, t) = U, zt(x, 0) = IJ(x),
where(12)are the errors of approximation on the solution of problerr1 (1)-(3) associ-
ated with equation (1) and the second initial condition (2), respectively.
If the coefficient k(x, t) and the right-hand side f(x, t) possess only a
finite number of immovable discontinuities, the grid wh = wh ( f{) will be so
chosen that all discontinuity lines will pass through the nodal points (the