Civil Engineering Reference
In-Depth Information
minimum‚ the numerical solution should obey these limiting conditions‚ and if
it does‚ we refer to the numerical integration formula as being stable.
Now consider the behavior of each of the above formulæ on this test equa-
tion‚ At the outset‚ we point out that since the step size
the sign of
is the same as that of
For the Forward Euler method under a
constant step size
so that
When
as
provided
Representing
the above equation may be rewritten as
In the this corresponds to a circle centered at (-1‚0) with a radius of
1‚ as shown in the left half plane (corresponding to in Figure 2.6(a)).
When satisfies the requirement that lies within this region‚ the resulting
solution satisfies the basic requirement of stability: that the asymptotic value
of the solution as time tends to
is the same as that for the exact solution.
On the other hand‚ for the case when we must have
as It is easily verified that as regardless of
what value of is chosen‚ implying that the test equation satisfies the stability
requirement for this case over the entire right half plane.
