Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

3.5 Numerical integration 62
3.5.5 Extrapolation methods (Bulirsch-Stoer)
The idea behind extrapolation methods is that the final answer for +1 is approximated
by an analytical function in the step size (Richardson extrapolation). The best strategy
is to use a rational extrapolating function as the ansatz, and not a polynomial function
(Bulirsch-Stoer). This function is determined and fitted to the problem of interest by
performing calculations with various values of . The extrapolating function is then
used for extrapolating from to +1 .
The optimal strategy for stiff DE systems is to follow the Bulirsch-Stoer method (Stoer
1980) in the form implemented by Deuflhard (Bader 1983, Deuflhard 1983, Deuflhard
1985). For Fortran subroutines, see (Press 1992).
I
Figure 3.4: The principle of extrapolation methods.
3.5.6 Example: Consecutive first-order reactions
As an example, we solve the DE’s for two consecutive first-order reactions
A 1
−→ B 2
−→ C (3.159)
(a)exactly,usingsymbolicalgebrasoftwarelikeMathCad,Maple,Mathematica,or
MuPad (the math engine built-in into SWP, the program used to write this script) and
(b) using numeric integration.