PHY 480 - Computational Physics
Michigan State University, Spring Semester 2006
DIFFERENTIAL EQUATIONS 1
Problem
d y
d x
= f(x, y), y(0) = y
0
Euler
y
n+1
= y
n
+ h f(x
n
,y
n
) + O(h
2
)
(1)
Mid-point predictor, trapezoidal corrector
p
n+1
= y
n
-
1
+ 2 h f(x
n
,y
n
) + O(h
3
)
y
n+1
= y
n
+
h
2
(f(x
n+1
,p
n+1
) + f(x
n
,y
n
)) + O(h
3
)
(2)
Taylor, third order
y
n+1
= y
n
+ h f
n
+
h
2
2
é
ê
ë
æ
ç
è
¶
f
¶
x
ö
÷
ø
n
+ f
n
æ
ç
è
¶
f
¶
y
ö
÷
ø
n
ù
ú
û
+ O(h
3
)
(3)
Two-step predictor-corrector
p
n+1
= y
n
+ h (
3
2
f(x
n
,y
n
)
-
1
2
f(x
n
-
1
,y
n
-
1
)) + O(h
3
)
y
n+1
= y
n
+
h
2
(f(x
n+1
,p
n+1
) + f(x
n
,y
n
)) + O(h
3
)
(4)
Adams-Bashforth four-step predictor
p
n+1
= y
n
+
h
24
(55 f
n
-
59 f
n
-
1
+37 f
n
-
2
-
9 f
n
-
3
) +
251
720
h
5
y
(5)
(
x
)
(5)
Adams-Moulton four-step corrector
y
n+1
= y
n
+
h
24
(9 f
n+1
+ 19 f
n
-
5 f
n
-
1
+f
n
-
2
)
-
19
720
h
5
y
(5)
(
x
)
(6)