The Initial Value Problem y' = f(x, y); y(c) = d 105
Convergence and Instability A numerical method for approximating
a solution to an initial value problem is said to be convergent if, assuming
t here is no round-off error, the numerical approximation approaches the exact
solution as the stepsize approaches zero. All the numerical methods presented
in this text are convergent. However, this does not mean that as the stepsize
approaches zero, the numeri cal approximation will always approach the exact
solution, since round-off error will always be present. Sometimes the error
of a numerical approximation turns out to be la rger than predicted by the
local discretization error estimate. And, furthermore as the stepsize is de-
creased, the error for a particular fixed value of the independent variable may
become larger instead of smaller. This phenomenon is known as numerical
instability. Numerical instability is a property of both the numerical method
and the initial value problem. That is, a numerical method may be unstable
for some initial value problems and stable for others. Numerical instability
usually arises because a first-order differential equation is approximated by
a second or higher order difference equation. The approximating difference
equation will have two or more solutions- the fundamental solution which
approximates the exact solut ion of the initial value problem, and one or more
parasitic solutions. The parasitic solutions are so named because they "feed"
upon the errors (both round-off and local discretization errors) of the numer-
ical approximation method. If the parasitic solutions remain "small" relative
to the fundamental solution, then the numerical method is stable; whereas,
if a parasitic solution becomes "large" relative to the fundamental solution,
then the numerical method is unstable. For h sufficiently small, single-step
methods do not exhibit any numerical instability for any initial value prob-
lems. On the other hand, multistep methods may be unstable for some initial
value problems for a particular range of values of the stepsize or for all step-
size. In practice, one chooses a particular numerical method and produces
a numerical approximation for two or more reasonably "small" stepsizes. If
the approximations produced are essentially the same, then the numerical
method is probably stable for the problem under consideration and the re-
sults are probably reasonably good also. If the results are not similar, then
one should reduce the stepsize further. If dissimilar results persist, then the
numerical method is probably unstable for the problem under consideration
and a different numerical method should be employed.
I EXAMPLE 7 Numerical Instability
Use Euler's method (Yn+l = Yn+hfn) and the midpoint rule (Yn+l = Yn-l +
2hfn) with a constant stepsize h = .1 to produce numerical approximations
on the interval [O, 2.4] to the initial value problem y' = -3y + l; y(O) = l.
Compare the numerical results with the exact solution.