174 Homogeneous Difference Sche1nes- A greater gain in accuracy on a sequence of grids. The Runge method.
In a common setting we are dealing with a linear mathematical-physics
equation
( 11) Lu=f(x), x E G,
regardless of additional conditions on the boundary r. Let w h be a grid in
the domain G with step h, by means of which a difference scheme(12)can be put in correspondence with problem (11).
When this scheme happens to be stable, so that llYhll(lh):::; Mll<f'hll( 2 h)
with a positive constant M independent of h (see Chapter 2, Section 2),
the approximation implies the convergence(13)where i/;h = <ph - Lh uh = (<ph - fh) - (Lh uh - (Lu)h) is the error of
approximation (residual) on the problem (11) solution u = u(x). It is
readily seen from (13) that the order of accuracy is not lower than the order
of approximation and the behaviour of the residual 111/Jh 11(2h) = O(hn) is a
corollary to the estimate for the relevant deviation II Yh - uh ll(ih) = O(hn).
In order to improve the accuracy of the approximate solution, it is
necessary to diminish the grid step h or make the approximation order
n of the scheme higher. An elementary example illustrating how to raise
the order of approximation on a solution is available in Chapter 1, Sec-
tion 2.2. However, in trying to develop higher-accuracy schemes for many
problems, especially for equations with variable coefficients, considerable
obstacles of technical character do arise. Moreover, the transition to such
schemes except the schemes for the heat conduction equation with constant
coefficients and the Laplace equation which will be given special investiga-
tion in Chapters 4~5, should cause the essential growth of the volume of
con1putations.
The gain in accuracy provided by refining the step h is limited by
requirements of econo111y. Such an approach is equivalent to minimizing the
execution ti111e necessary in this connection in obtaining the solution. But
if the solution of the original problem u and f both are smooth functions
of x, the accuracy of numerical solution can be increased by performing
calculations for the same problem (12) on a sequence of grids wh,, ... , whn.
In the sequel we assume that u = u( x) possesses all necessary deriva-
tives which do arise in the further development. In order to understand