102 Ordinary Differential Equationsc. How small must the stepsize be to ensure six decimal place accuracy
per step?2. a. Use the midpoint rule and a stepsize h = .1 to compute a numerical
approximation to the initial value problem y' = x^2 - y; y(O) = 1
on the interval [O, l]. Use the exact solution values for starting
values.b. Estimate the maximum local discretization error on the interval
[O, 1].
c. How small must the stepsize be to ensure six decimal place accuracy
per step?- Consider the initial value problem y' = x^2 - y; y(O) = l.
a. For the given initial value problem solve the Adams-Moulton for-
mula for m = 0 expli citly for Yn+l ·
b. Use the formula derived in part a. to produce a numerical solution
to the given initial value problem on the interval [O, 1] using a
stepsize of h = .1.
- Make a table of values and compare the numerical approximations
produced in exercises 1-3 with the exact solution of the initial value
problem y' = x^2 - y; y(O) = 1 on the interval [O, l].
2.4.3 Predictor-Corrector Methods
The Adams-Moulton formulas of the previous section are implicit formulas.
In general, f(x, y) will be nonlinear and it will be impossible to solve the
implicit formula expli citly for Yn+l · However, we can try to determine Yn+l
by iteration. That is, we obtain, in some manner, a first approximation to
Yn+1, call it y~j', and then we successively calculate J;;:.? = f(xn+1, v;;!i)
and use this approximation of f n+l in the implicit formula to successively
calculate y;;~t^1 > for k = 0, 1, .... Under fairly general conditions on the
function f, it can be shown that for sufficiently small values of the stepsize h,
the sequence < v;;!{' >k°=l converges to a solution Yn+l of the implicit formula
and that the solut ion Yn+l is unique. [Note that Yn+l will be the solution of the
implicit formula; but, in general, it will not be the solution of the differential
equation at Xn+i, ¢(xn+ 1 ).] An explicit formula that is used to obtain the
first approximation, y;;~{, is called a predictor formula and an implicit
formula used in t he iteration procedure to calculate v;;!? for k = 1, 2, ... is
called a corrector formula. One usually chooses the predictor and corrector
formulas of the iteration procedure so that the order of the local discretization
error of each formula is nearly the same. Generally, the corrector formula is
chosen so that the local error is smaller than the lo cal error of the predictor
formula. Both single-step and multistep predictor-corrector methods may be
devised.