Mathematical Methods for Physicists: A concise introduction - Site Map

NUMERICAL METHODS
Runge±Kutta method is that we do not need to compute partial derivatives, but it
becomes rather complicated if pursued for more than two or three steps.
The accuracy of the Runge±Kutta method can be improved with the following
formula:
y n‡1 ˆ y n ‡ h 6 …k 1 ‡ 2k 2 ‡ 2k 3 ‡ k 4 †;
…13:25†
where
k 1 ˆ f …x n ; y n †;
k 2 ˆ f …x n ‡ h=2; y n ‡ hk 1 =2†;
k 3 ˆ f …x n ‡ h; y 0 ‡ hk 2 =2†;
k 4 ˆ f …x n ‡ h; y n ‡ hk 3 †:
…13:25a†
…13:25b†
…13:25c†
…13:25d†
With this formula the error in y n‡1 is of order h 5 .
You may wonder how these formulas are established. To this end, let us go
back to Eq. (13.22), the three-term Taylor series, and rewrite it in the form
where
y 1 ˆ y 0 ‡ hf 0 ‡…1=2†h 2 …A 0 ‡ f 0 B 0 †‡…1=6†h 3 …C 0 ‡ 2f 0 D 0 ‡ f 2 0 E 0
‡ A 0 B 0 ‡ f 0 B 2 0†‡O…h 4 †; …13:26†
A ˆ @f
@x ;
B ˆ @f
@y ;
C ˆ @2 f
@x 2 ;
D ˆ @2 f
@x@y ;
E ˆ @2 f
@y 2
and the subscript 0 denotes the values of these quantities at …x 0 ; y 0 †.
Now let us expand k 1 ; k 2 ,andk 3 in the Runge±Kutta formula (13.24) in powers
of h in a similar manner:
Thus
k 2 ˆ f …x 0 ‡ h=2; y 0 ‡ k 1 h=2†;
k 1 ˆ hf …x 0 ; y 0 †;
ˆ f 0 ‡ 1 2 h…A 0 ‡ f 0 B 0 †‡ 1 8 h2 …C 0 ‡ 2f 0 D 0 ‡ f 2 0 E 0 †‡O…h 3 †:
2k 2 k 1 ˆ f 0 ‡ h…A 0 ‡ f 0 B 0 †‡
and
d
dh …2k 2 k 1 † ˆ f 0 ;
hˆ0
!
d 2
dh 2 …2k 2 k 1 †
hˆ0
ˆ 2…A 0 ‡ f 0 B 0 †:
474