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

3.5 Numerical integration 58
I
Survey of methods of numerical integration methods:
• explicit methods (e.g., Runge-Kutta),
• implicit methods (e.g., Gear),
• extrapolation methods (e.g., Richardson, Bulirsch-Stoer, Deuflhard).
In the following, we consider the basics of the most convenient methods.
3.5.1 Taylor series expansions
The Taylor expansion is the basis for all numerical integration methods. We start from
the initial value of ( 0 ) at some time 0
( 0 )= 0 (3.136)
and want to determine the value ( 0 + ) at time 0 + . UsingtheDEfor
we expand in a Taylor series around 0 :
Recursion formula:
·
() =( ) (3.137)
( 0 + ) = ( 0 )+ · · ( 0 0 )+ 2
2! · ··
( 0 0 )+ (3.138)
= ( 0 )+ · ( 0 0 )+ 2
2! · ·
( 0 0 )+ (3.139)
+1 = + · ( )+ 2
2! · ·
( )+ (3.140)
The first-order term ( ) is given by the rate equation at , the second-order
term ( ·
) (which is taken into account for example by the 2 order Runge-Kutta
method) and higher order terms (⇒ higher order Runge-Kutta methods) have to be
evaluated numerically by finite differences.
3.5.2 Euler method
The Euler method is based on a 1 order Taylor expansion. If the step size is small
enough, the higher order terms can be neglected:
( 0 + ) = ( 0 )+ · ·
( 0 )+O( 2 ) (3.141)
Initial value problem:
·
= ( ) (3.142)