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

3.5 Numerical integration 61
I 4 order Runge-Kutta method: In practice, one frequently uses the 4 order
Runge-Kutta method.
I
Figure 3.3: Illustration of the 4 order Runge-Kutta method.
3.5.4 Implicit methods (Gear):
The Runge-Kutta method becomes unstable for stiff DE systems. In this case, it is
preferable to employ implicit (also called iterative or backward) integration methods.
The starting point is the implicit form of the Euler method,
( 0 + ) = ( 0 )+ · ·
( 0 + )+O( 2 ) (3.156)
For the initial value problem ·
= ( ) with ( 0 )= 0 this leads to the recursion
formula:
+1 = + (3.157)
+1 = + ( +1 +1 ) (3.158)
Using the so-called predictor-corrector methods, +1 is predicted in a first step by
starting from using one of the explicit methods described above. The obtained trial
value (0)
+1 is then corrected by using a backward integration scheme (Gear 1971a, Gear
1971a, Press 1992) to obtain a better estimate +1. (1) Then, and (1)
+1 are used for
a new prediction-corrrection cycle, giving +1. (2) The procedure is repeated until the
difference between successive iterations falls below some tolerance limit .
The predictor-corrector method of Gear has been the method of choice for a long time
(Gear 1971a, Gear 1971a).