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A part of the ce 341/441 lecture notes from fall 2004. It covers the solutions to ordinary differential equations (odes) and the implementation of runge-kutta methods. How to find the solution of an ode with initial conditions and introduces the 2nd order runge-kutta methods. The document also discusses the approximation error and the comparison of different methods.
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CE 341/441 - Lecture 21 - Fall 2004
p. 21.
INSERT FIGURE NO. 96
dy ----- dt
f^
y t ,(
y t
o (^
y t ( )
t^
f^
y t ,(
y^
t
y(t)
y^0
t
f(y,t)=slope
CE 341/441 - Lecture 21 - Fall 2004
p. 21.
y^
j^
1 +^
y^
j^
t Φ
=
a
g 1 1
a
g 2
2
a
1
f^
t^ j
y^
j , (^
a^2
f^
t^ j
p
1
t y
j^
p
2
t f
t^ j
y j , (^
t^ j
y j , (^
f^
t^ j
p
t y
j ,^
p
t f
t^ j
y j , (^
f^
t^ j
y j , (^
t^ j^
y^ , i (^
)
y
f ∂
t^ j^
y^ , j (^
)
2
f ∂
y (^
2
f ∂
y ∂
t
y (^
2
f ∂
t^ j^
y^ , j (^
)^
p
t
y^
p^2
t f
t^ j
y^
j , (^
CE 341/441 - Lecture 21 - Fall 2004
p. 21.
⇒
⇒
y^
j^
1 +^
y t
j^
1
(^
t^ j
y^
j^
1 +^
y^
j^
dy ----- tdt
j
t (^
d -
j
t (^
dy ----- dt
f^
t y ,(
d
f ∂
dy ----- dt
d
∂ f^ ∂
f ∂
f
CE 341/441 - Lecture 21 - Fall 2004
p. 21.
⇒
dy ----- dt
j
d
j
y^
j^
1
y^
j^
1 +^
t (^
y^
j^
1 +^
y^
j^
t f
j
t (^
j
f ∂
f j j
t (^
y^
j^
1
y^
j^
1 +^
y^
j^
t a
1
a^2
(^
f^
j^
t (^
a
2
p
1
j
a
2
p^2
f^
j
f ∂
j
t (^
a
1
a
2
a^2
p
1
a^2
p
2
a^1
a
2
=
p
1
p
2
1 2 a
CE 341/441 - Lecture 21 - Fall 2004
p. 21.
→
yj+
tj+
tj
t
t
yj
fj
avg.slope
slope = f(t
+j
t, y
+j
t f
)j
avg. slopeslope = f(t
, yj
)j
y^
j^
1
t^ j
1 +^
y^
j^
1 +^
f^
t^ j
t y
j^
t f
j
y^
j^
1 +^
t^ j
y j , (^
t^ j
1
y^
j^
1
CE 341/441 - Lecture 21 - Fall 2004
p. 21.
⇒
←
←
←
a^
a^1
a
2
p
1
p
2
y^
j^
1 +^
y^
j^
t f
t^ j
t 2 ----
y^
j
t 2 ----
f^
j
t^ j
1 --- 2 +^
t^ j
=
y^
j^
1 --- 2 *****^ +
y^
j
f
t^ j
y j , (^
y^
j^
1 +^
y^
j^
t f
t^ j
1 --- 2 +^
y^
j^
1 --- 2 ***** + ,
f^
t y ,(
CE 341/441 - Lecture 21 - Fall 2004
p. 21.
y^
j^
1 +^
y^
j^
t
g^1 ----^6
g
g
g
y^
j^
1 --- 2 *^ +
y^
j
f
y^
j^
t^ , j
(^
y^
j^
1 --- 2 **^ +
y^
j
f
y^
j^
1 --- 2 *^ +
t^ j
1 --- 2
,
y^
j^
1 *^ +
y^
j^
t f
y^
j^
1 --- 2 **^ +
t^ j
1 --- 2
,
y^
j^
1 +^
y^
j^
t^
f^
y^
j^
t^ , j
(^
f^
y^
j^
**1 --- 2
***^
t^ j
1 --- 2 + ,
f^
y^
j^
**1 --- 2
t^ j
1 --- 2 + ,
f^
y^
j^
**1 ***^ +
t^ j
1 + ,
CE 341/441 - Lecture 21 - Fall 2004
p. 21.
→
yj+1/
tj+
=
tj
t
yj
t+j
∆ t^2
tj^
+
∆ t
**j+1/2y *
*yj+
use further improved value of slope to obtain y
j+
3
f(y
j + 1/
,^ t
j + 1/
)
3
use improved slope to evaluate the new midpoint y
**j+1/
2
f(y
j + 1/
,^ t
j + 1/
)
2
f(t
+j y
) used to estimatej
y *^ j+1/
1
y^
j^
1
CE 341/441 - Lecture 21 - Fall 2004
p. 21.
→
⇒
⇒
j^
1
y ˆ^
j^
1 +^
y^
j^
1
k –
y ˆ^
j^
1 +^
t /
y^
j^
1 +^
t
k
