PHY 480 - Computational Physics
Michigan State University, Spring Semester 2006
DIFFERENTIAL EQUATIONS 2: Runge-Kutta
Second order
k
1
= h f(x
n
,y
n
)
y
n+1
= y
n
+ h f(x
n
+1/2h,y
n
+1/2k
1
) + O(h
3
)
(1)
Third order
k
1
= h f(x
n
,y
n
)
k
2
= h f(x
n
+1/2h,y
n
+1/2k
1
)
k
3
= h f(x
n
+h, y
n
-
k
1
+ 2 k
2
)
y
n+1
= y
n
+
1
6
(k
1
+ 4 k
2
+ k
3
) + O(h
4
)
(2)
Fourth order
k
1
= h f(x
n
,y
n
)
k
2
= h f(x
n
+1/2h,y
n
+1/2k
1
)
k
3
= h f(x
n
+1/2h,y
n
+1/2k
2
)
k
4
= h f(x
n
+h, y
n
+ k
3
)
y
n+1
= y
n
+
1
6
(k
1
+ 2 k
2
+ 2 k
3
+ k
4
) + O(h
5
)
(3)
Adaptive stepsize control
2h: y(x+2h) = y
1
+ (2 h)
5
f
+ O(h
6
)
h: y(x+2h) = y
2
+ 2 (h
5
)
f
+ O(h
6
)
D
= y
2
-
y
1
(4)
y(x+2 h) = y
2
+
D
15
+ O(h
6
)
(5)