Hybrid Methods for Initial Value Problems in Ordinary Differential ...

HYBRID METHODS FOR INITIAL VALUE PROBLEMS
Positive al would improve the stability slightly since it lends to increase
a10and all froni negative values, and to decrease the other al, from positive
values. However, positive al will increase the predictor error, and thus a
large error will result if ILX is not zero.
6. Conclusion. The most obvious conclusion that can be drawn fro111
Tables 3 and 4 is that one should not use Runge-Kutta if high order derivatives
are well behaved! For the general problem, it is difficult to say
much more. If the evaluation of the derivative is time consuming, then it
seems reasonable to compare Adanis' single corrector h = A,and the other
lliethods (including Adan~s double corrector) for h = &. Each of these
niethods taltes 32 evaluations per unit step in x. Kordsiecli's inetllod differs
froni Adams' niethod only in the predictor and in the effect ~vheii the step
size is changed, but this apparently introduces a co~lsicierable ai~iount of
error. Automatic step size changing probably would pay off in cascs where
there is a sudden change of behavior in the function; for those cases one
expects Nordsieck's method to be superior. A step changing niechanislii
can be programmed on the basis of the difference between the predictor and
corrector of the hybrid methods presented. Although this is a term of order
h(2kt+ (2A-2' + 0(hZN3),it is indicative of the behavior of the '.average"
y
value of the error term which includes ternls in h2k+3y'2h-3) (a) and
h2k+3 Xy(2a-2)
(2).
The hybrid 6th order has about three times the error of Adams' double
corrector in one case, and about 4 in the other. It has the advantage of
using two additional points instead of 4 (5 if Adams-Bashforth is used as a
corrector). If storage space and hence the number of points is a criterion
in large vector problems, then the 8th order hybrid method is iiiore accurate
but still requires fewer additional points. If one is prepared to use more
function cvaluations and/or use other points besides the half point, prob-
ably higher order stable methods exist and would pay off in special cases;
but it seems unlikely that one would want to go beyond order 8 or 10, as it
appears to become harder to control the larger number of extraneous roots,
or to have much knowledge of the high order derivatives.
Gragg and Stetter [2]consider correctors sinlilar to the corrector used
in this paper with J = 2. They choose the point of evaluation of the first
predictor to optimize the (stable) order of the corrector. This introduces
an additional degree of freedoin not existing in the correctors used here,
allowing a corrector of order 2k + 4 rather than 21c + 2 to be chosen. Their
surprising result is that for k 5 3 (the k of Gragg and Stetter is one larger
than this one), these optimal order methods are stable. The result for 11 2 4
is not yet known. Because their corrector order is greater, the predictors
must be of a correspondingly higher ordsr, meaning that additional inesh
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