Analysis of the ODE Integration Methods
June 9, 1996
5.0 Analysis of the ODE Integration Methods
According to many books on numerical analysis, the multi-step Adam-Bashford
method shouldn't even be in the running for the best method, and my taylor3 method
should be a ignored. After all, the Adam-Bashford method isn't even as sophisticated as a
predictor-corrector method and the Taylor series is rarely talked about except for its use as
a fundamental theory.
A typical example of these opinions can be found in the highly regarded
cal Recipes in C
which has these comments on the subject:
Runge-Kutta succeeds virtually always; but it is not
usually fastest. Predictor-corrector methods, since they use
past information, are somewhat more difficult to start up,
but, for many smooth problems, they are computationally
more efficient than Runge-Kutta. In recent years Bulirsch-
Stoer has been replacing predictor-corrector in many appli-
cations, ... it appears that only rather sophisticated predictor-
corrector routines are competitive.(14:568)
[ The straight Adam-Bashford method can hardly be considered a "sophisticated"
method... ]
The techniques described in this section [Bulirsch-
Stoer] are not for differential equations containing non-
smooth functions. ... A second warning is that the tech-
niques in this section are not particularly good for differen-
tial equations which have singular points
the interval
of integration.(14:582)
We suspect that predictor-corrector integrators have
had their day, and that they are no longer the method of
choice for most problems in ODEs. For high-precision
applications, or applications where evaluations of the right
hand sides are expensive, Bulirsch-Stoer dominates. For
convenience, or for low-precision, adaptive-step size
Runge-Kutta dominates. Predictor-corrector methods have
been, we think, squeezed out in the middle. There is possi-
bly only one exceptional case: high-precision solution of
very smooth equations with very complicated right-hand
sides, as we will describe later.(14:589)
Previous page    Home    Table of Contents    Next page
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57
This document is best viewed as n-body.pdf because the translation to html was buggy.