178 Homogeneous Difference Schemeson the behaviour of the solution of an original problern is available, a grid
can be made so as to achieve a prescribed accuracy at minimum nodal
points and, therefore, in a minimal number of the necessary operations.
For instance, successive grid refinement will be appreciated in the region of
widely varying coefficients and the right-hand side. Specifically, near the
point (the line) of discontinuity of coefficients, representing the boundary
between two media, the grid is refined in such a way to attain the minirnum
of the step near the boundary. After that, the step is being enlarged for
instance, as a geometric progression in moving from the boundary. When no
information about the behaviour of the solution is available, a preliminary
cornputation proves to be useful on a sparse grid, after which we are working
on a new grid with a more smaller step in the regions with large deviations
of the solution. In practice non-equidistant grids are widely used. As
shown in Section 3, it is possible to construct special grids wh(J{) so that
all discontinuity points of the coefficient k(x) involved in the equation(k(x) u^1 )1- q(x) u = -f(x)
fall into nodal points of the grid wh(J{). Under such a choice, any ho-
mogeneous difference scheme ( ayx) i: - dy = -tp generating approximation
of order 2 (in the class of smooth coefficients) is of second-order accuracy
in the class of discontinuity coefficients k( x). The accuracy of difference
schemes, which depends on the existing grids suitable to computer calcu-
lations for equations with variable coefficients, needs investigation for each
concrete problem.
3.5 OTHER PROBLEMS- The third boundary-value problem. The main goal of our studies is
a homogeneous difference scheme for the boundary-value problem of the
third kind:
(1)
, Lu= (k(x)u^1 )
1- q(x)u = -f(x),
O<x<l, k(x)>c 1 >0, q>O,
k(O) u'(O) = {3 1 u(O) - P 1 , -k(l) u'(l) = {3 2 u(l) - 1! 2 ,
f31 + f32 > 0.
We approximate equation (1) in the usual way:
(2) Ay = -<p(x), Ay = (ay 5 ;)"' - dy, a 2 c 1 > 0, d 2 0,