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Iterative solution of algebraic equations puts the nonlinear term on the r.h.s. and iterates until convergence. There may be convergence problems. Approximation of Ordinary Differential Equations, Initial Value Problems, Boundary Value Problems, Initial Value Problems, Euler's Method, Time Stepping, Time Marching
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CE 341/441 - Lecture 15 - Fall 2004
p. 15.
→
→
-^
→
d
dy ----- dt
Cy
g t
y^
dy ----- dt
o
=
y t
CE 341/441 - Lecture 15 - Fall 2004
p. 15.
d
2 y d x
dy ----- dx
Ey
h x
y^
y L (
d
dy ----- dt
g t
y^
dy ----- dt
z^
dy ----- dt
dy ----- dt
z =
y^
y^ o =
dz ----- dt
z^
g t
z^
v^ o =
CE 341/441 - Lecture 15 - Fall 2004
p. 15.
⇒
⇒
dy ----- dt
f^
y t ,(
y t
o (^
j^
dy ----- dt
j
dy ----- dt
j
f^
y^
, j t^ j (^
y^
j^
1 +^
y^
j
t
f^
y^
j^
t^ , j
(^
y^
j^
1 +^
y^
j^
t f
y^
j^
t^ , j
(^
CE 341/441 - Lecture 15 - Fall 2004
p. 15.
- Simply “march” forward in time - From time level
j^
t^ j
j^
t^ j
1 +^
t^ j
t
yj+
yj
tj^
tj+
∆
t f (t
, yj
)j
f^
t^ j
y^
j , (^
t^ j
y^
j^
1 +^
t f
t^ j
y j , (^
y^
j
CE 341/441 - Lecture 15 - Fall 2004
p. 15.
⇒
i dy ----- dt
i
y^ i
y i
=
dy ----- dt
i
y^ i
1 +^
y^ i
t
y^ i
y i
=
y^ i
1 +^
y^ i
t y
i^
y^ i
i^
i^
CE 341/441 - Lecture 15 - Fall 2004
p. 15.
⇒
y^ o
= i
i^
t^ i
i^
t^
t^ o
⋅^
i ∆
t
t^ o
y^1
y^ o
ty
o^
y^ o
y^1
y^1
t^1
i^
i^
y^2
y^1
t y
1
y^1
y^2
y^2
t^2
CE 341/441 - Lecture 15 - Fall 2004
p. 15.
y^5
t^5
y^6
t^6
t^
t^
CE 341/441 - Lecture 15 - Fall 2004
p. 15.
→
→
t
t^
y^
j^
1
CE 341/441 - Lecture 15 - Fall 2004
p. 15.
... ...
n^
n^
y^0
y^2
y^1
-^
y^0
x (^
y
1
x (^
y^3
y^2
-^
y^1
x (^
y
2
x (^
y^
j^
1 +^
y^
j
-^
y^
j^
1
-^
x (^
y^
j
x (^
y^ n
y^ n
1
-^
y^ n
2
x (^
y^
n^
1
x (^
y^ n
1 +^
y^ n
-^
y^ n
1
-^
x (^
y^
n
x (^
y^ n
1 +^
CE 341/441 - Lecture 15 - Fall 2004
p. 15.
α
α
α
α
α
α
y^0 y^1 y^2 y^3 y^ n
1
1
x (^
x (^
x (^
x (^
x (^
α
x (^
CE 341/441 - Lecture 15 - Fall 2004
p. 15.