Difference approximation of ele1nentary differential operators 87into the formulau(x, r) - u ou I T2 a2u I
0 (x) = r - 0 + - 2 - 0 2 + O(r(^3) )
i t=O t t=O
yieldsμ= L u 0 + f(x, 0), so that
y(x, r) = u 0 (x) + T (u~(x) + J(x, 0)).
2.3 STABILITY OF A DIFFERENCE SCHEME
- Examples of stable and unstable difference schen1es. As we have shown
in the preceding section, the introduction of a difference scheme permits one
to reduce the solution of a problem associated with a differential equation to
a system of linear algebraic equations. In this situation the right-hand sides
of equations, boundary and initial conditions, all of which we will call in the
sequel the input data, are specified with a certain error. This procedure
causes rounding errors to be inevitable in the process of numerical solution
of a system. Because of this, it seems natural to require that small errors
in specifying the input data should not increase during the process of the
execution and result in wrong reasoning. The schemes in which initial errors
grow during the course of calculations turn out to be unstable and, from
the viewpoint of possible applications, cannot find response.
We will not attempt to encompass a wide variety of situations, but in-
stead look in more detail at several exhaustive examples before formulating
the definition of stability of a difference schen1e with respect to the input
data, the concept of which we have intuitively developed earlier.
Example 1. A stable schem.e. Let
(1) u' = -CY u' x > 0' u(O) = Uo ' C\' > 0.
It is straightforward to verify that the function u( x) = u 0 exp {-CY x}
gives the exact solution of problem (1). This solution does not increase with
increasing x: I u( x) I < I u 0 I for CY > 0, so that u( x) continuously depends
on u 0. An excellent start in this direction is to approximate problem ( 1) on
the equidistant grid wh = {xi = ih, i = 0, 1, ... } by the difference problem
Yi - Yi-1
(2) h +CY Yi= 0' Yo = Uo, i = 1, 2, ... '