Analysis of the ODE Integration Methods
June 9, 1996
51
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
Numeri-
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
inside
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)
...
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