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

3.5 Numerical integration 60
2 order Runge-Kutta ansatz = Taylor expansion to 2 order:
( 0 + ) = ( 0 )+ · ( 0 )+ 2 ··
2! ( 0 )+O( 3 ) (3.149)
Ã
≈ ( 0 )+ ·
·
( 0 )+ 2 ( 0 + ) − ·
!
( 0 )
+ O( 3 ) (3.150)
2
= ( 0 )+ ³ ·(0 + )+ ·( 0 )´
2 | {z }
=2× slope of () at midpoint
+ O( 3 ) (3.151)
Recursion formula:
+1 = + (3.152)
+1 = + 2 + O( 3 ) (3.153)
1 = ( ) (3.154)
2
µ
=
+ 2 + 1
2
(3.155)
I
Figure 3.2: Illustration of the 2 order Runge-Kutta method.