Introduction to Simulation - Lecture 12
Methods for Ordinary Differential Equations
Jacob White
Thanks to Deepak Ramaswamy, Jaime Peraire, Michal
Rewienski, and Karen Veroy
Outline
Initial Value problem examples
Signal propagation (circuits with capacitors).
Space frame dynamics (struts and masses).
Chemical reaction dynamics.
Investigate the simple finite-difference methods
Forward-Euler, Backward-Euler, Trap Rule.
Look at the approximations and algorithms
Examine properties experimentally.
Analyze Convergence for Forward-Euler
Application
Problems
Signal Transmission in an
Integrated Circuit
Signal Wire
Wire has resistance
Wire and ground plane form a capacitor
Logic
Gate
Ground Plane
• Metal Wires carry signals from gate to gate.
• How long is the signal delayed?
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Logic
Gate
Application
Problems
Signal Transmission in an
Integrated Circuit
Circuit Model
resistor
capacitor
Constructing the Model
• Cut the wire into sections.
• Model wire resistance with resistors.
• Model wire-plane capacitance with capacitors.
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Application
Problems
Oscillations in a Space
Frame
• What is the oscillation amplitude?
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Application
Problems
Oscillations in a Space
Frame
Simplified Structure
Bolts
Struts
Ground
Example Simplified for Illustration
Load
Application
Problems
Oscillations in a Space
Frame
Modeling with Struts, Joints and
Point Masses
Point Mass
Strut
Constructing the Model
• Replace Metal Beams with Struts.
• Replace cargo with point mass.
Application
Problems
Chemical Reaction
Dynamics
Crucible
Reagent
Strange green
stuff
• How fast is product produced?
• Does it explode?
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Signal Transmission in an
Integrated Circuit
Application
Problems
v1
iC1
C1
iR1
A 2x2 Example
v2
iR2
R2
R1 R3
iR3
iC2
C2
Constitutive
Equations
dv
ic = C c
dt
1
iR = vR
R
Conservation
Laws
iC1 + iR1 + iR2 = 0
iC2 + iR3 − iR2 = 0
Nodal Equations Yields 2x2 System
⎡C1
⎢0
⎣
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1
⎡1
⎡ dv1 ⎤
+
⎢
R1 R2
0 ⎤ ⎢ dt ⎥
⎢
⎥ = −⎢
⎥
C2 ⎦ ⎢ dv2 ⎥
1
⎢
⎢ −R
⎢⎣ dt ⎥⎦
2
⎣
1 ⎤
−
R2 ⎥ ⎡ v1 ⎤
⎥⎢ ⎥
1
1 ⎥ ⎣ v2 ⎦
+ ⎥
R3 R2 ⎦
Application
Problems
Signal Transmission in an
Integrated Circuit
A 2x2 Example
Let C1 = C2 = 1, R1 = R3 = 10, R2 = 1
⎡C1
⎢0
⎣
1
⎡1
⎡ dv1 ⎤
+
⎢R R
0 ⎤ ⎢ dt ⎥
1
2
⎢
⎥ = −⎢
⎥
C2 ⎦ ⎢ dv2 ⎥
1
⎢
−
⎢ R
⎢⎣ dt ⎥⎦
2
⎣
dx ⎡ −1.1 1.0 ⎤
=⎢
x
⎥
dt ⎣ 1.0 −1.1⎦
A
1 ⎤
R2 ⎥ ⎡ v1 ⎤
⎥
1
1 ⎥ ⎢⎣ v2 ⎥⎦
+
R3 R2 ⎥⎦
−
Eigenvalues and Eigenvectors
⎡1 −1⎤ ⎡ −0.1 0 ⎤ ⎛ ⎡1 −1⎤ ⎞
A=⎢
⎜⎢
⎟
⎥
⎢
⎥
⎥
−2.1⎦ ⎝ ⎣1 1 ⎦ ⎠
⎣1 1 ⎦ ⎣ 0
eigenvectors
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Eigenvalues
−1
An Aside on Eigenanalysis
dx(t )
Consider an ODE:
= Ax(t ),
dt
Eigendecomposition:
x(0) = x0
#
# ⎤ ⎡ λ1 0 0 ⎤ ⎡ #
⎡#
A = ⎢⎢ E1 E2 En ⎥⎥ ⎢⎢ 0 % 0 ⎥⎥ ⎢⎢ E1
⎢⎣ #
#
# ⎥⎦ ⎢⎣ 0 0 λn ⎥⎦ ⎢⎣ #
E
#
E2
#
Change of variables: Ey (t ) = x (t ) ⇔ y (t ) = E −1 x (t )
dEy (t )
= AEy (t ), Ey (0) = x0
Substituting:
dt
⎡λ1 0 0 ⎤
⎢
⎥
−1 dy (t )
−1
Multiply by E :
= E AEy (t ) = ⎢ 0 % 0 ⎥ y(t )
⎢0 0 λ ⎥
dt
n ⎦⎥
⎣⎢
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# ⎤
En ⎥⎥
# ⎥⎦
−1
An Aside on Eigenanalysis Continued
From last slide:
⎡λ1 0 0 ⎤
dy(t ) = ⎢ 0 % 0 ⎥
dt ⎢⎢ 0 0 λ ⎥⎥
n⎦
⎣
y(t )
Decoupled
Equations!
dyi (t )
λi t
= λi yi (t ) ⇒ yi (t ) = e y (0)
Decoupling:
dt
dx(t )
Steps for solving
= Ax(t ), x(0) = x0
dt
1) Determine E , λ
λ1t
−1
⎡
e
0
0 ⎤
2) Compute y (0) = E x0
⎢
⎥
3) Compute y (t ) = ⎢ 0 % 0 ⎥ y (0)
⎢ 0 0 eλnt ⎥
⎣
⎦
4) x (t ) = Ey (t )
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Application
Problems
Signal Transmission in an
Integrated Circuit
A 2x2 Example
v1 (0) = 1
v2 (0) = 0
Notice two time scale behavior
• v1 and v2 come together quickly (fast eigenmode).
• v1 and v2 decay to zero slowly (slow eigenmode).
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Application
Problems
y = y0 + u
fs
fm
Struts, Joints and point
mass example
A 2x2 Example
Constitutive
Equations
d 2u
fm = M 2
dt
Conservation
Law
fs + fm = 0
y − y0 EAc
fs = EAc ∗
u
=
y0
y0
Define v as velocity (du/dt) to yield a 2x2 System
⎡M
⎢0
⎣
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⎡ dv ⎤ ⎡
EAc ⎤
⎢
⎥
0 −
0 ⎤ dt
⎡v ⎤
⎢
⎥
y0 ⎢ ⎥
⎢ ⎥=⎢
⎥
⎥ ⎣u ⎦
1 ⎦ ⎢ du ⎥
0 ⎥⎦
⎢⎣ dt ⎥⎦ ⎢⎣1
Application
Problems
Struts, Joints and point
mass example
A 2x2 Example
EAc
=1
Let M = 1,
y0
⎡M
⎢0
⎣
⎡ dv ⎤ ⎡
EAc ⎤
⎢
⎥
−
0
0 ⎤ dt
⎡v ⎤
⎢
⎥
y
=
⎢ ⎥
0
⎥ ⎢⎣u ⎥⎦
1 ⎥⎦ ⎢ du ⎥ ⎢
0 ⎥⎦
⎢⎣ dt ⎥⎦ ⎢⎣1
dx ⎡ 0 −1.0 ⎤
=⎢
x
⎥
0 ⎦
dt ⎣1.0
A
Eigenvalues and Eigenvectors
⎡ −1 −1⎤ ⎡ i 0 ⎤ ⎛ ⎡ −1 −1⎤ ⎞
A=⎢
⎜⎢
⎟
⎥
⎢
⎥
⎥
⎣ i −i ⎦ ⎣0 −i ⎦ ⎝ ⎣ i −i ⎦ ⎠
eigenvectors
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Eigenvalues
−1
Application
Problems
Struts, Joints and point
mass example
A 2x2 Example
1
v (0) = 1
0.5
0
u (0) = 0
-0.5
-1
0
5
10
Note the system has imaginary eigenvalues
• Persistent Oscillation
• Velocity, v, peaks when displacement, u, is zero.
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15
Application
Problems
Chemical Reaction
Example
A 2x2 Example
Amount of reactant = R, the temperature = T
dT
= −T + R
dt
dR
= − R + 4T
dt
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More reactant causes temperature to rise,
higher temperatures increases heat dissipation
causing temperature to fall
Higher temperatures raises reaction rates,
increased reactant interferes with reaction
and slows rate.
Application
Problems
Chemical Reaction
Example
A 2x2 Example
⎡ dT ⎤
⎢ dt ⎥ ⎡ −1 1 ⎤ ⎡T ⎤
⎢ ⎥=⎢
⎥ ⎢ R⎥
−
dR
4
1
⎦⎣ ⎦
⎢ ⎥ ⎣
⎢⎣ dt ⎥⎦
dx ⎡ −1 1 ⎤
=⎢
x
⎥
dt ⎣ 4 −1⎦
A
Eigenvalues and Eigenvectors
⎡ 1 −1⎤ ⎡1 0 ⎤ ⎛ ⎡ 1 −1⎤ ⎞
A=⎢
⎥ ⎢0 −3⎥ ⎜ ⎢ 2 2 ⎥ ⎟
2
2
⎣
⎦⎣
⎦⎝⎣
⎦⎠
eigenvectors
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Eigenvalues
−1
Chemical Reaction
Example
Application
Problems
A 2x2 Example
12
10
8
6
R (0) = 0
4
T (0) = 1
2
0
0
0.5
1
1.5
2
Note the system has a positive eigenvalue
• Solutions grow exponentially with time.
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2.5
Basic Concepts
Finite Difference
Methods
∆t ∆t
First - Discretize Time
0
t1
∆t
Second - Represent x(t) using values at ti
3
x̂ 2
4
x̂
1
x̂
x̂
0
t1
t2
Third - Approximate
t3
tL
d
the
)
x(tlusing
dt
l
l −1
d
xˆ − xˆ
Example:
x (tl ) dt
∆tl
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t L −1 t L = T
t2
l
xˆ x(tl )
Approx.
sol’n
discrete
xˆ l +1 − xˆ l
or
∆tl +1
Exact
sol’n
xˆ l 's
Finite Difference
Methods
x
slope =
d
x(tl )
dt
∆
slope =
tl
tl +1
x(tl +1 ) − x(tl )
∆t
Basic Concepts
Forward Euler Approximation
x(tl +1 ) − x(tl )
d
x(tl ) = A x(tl ) ≅
∆t
dt
or
x(tl +1 ) ≅ x(tl ) + ∆t A x(tl )
t
∆ = x(tl +1 ) − ( x(tl ) + ∆t A x(tl ) )
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Finite Difference
Methods
Basic Concepts
Forward Euler Algorithm
x(t1 ) xˆ1 = x(0) + ∆tAx ( 0 )
x
x(t2 ) xˆ 2 = xˆ1 + ∆tAxˆ1
#
x(t L ) xˆ L = xˆ L −1 + ∆tAxˆ L −1
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∆tAxˆ1
∆tAx (0)
t1
t2
t3
t
Basic Concepts
Finite Difference
Methods
x
slope =
∆
d
x(tl +1 )
dt
slope =
tl
tl +1
x(tl +1 ) − x(tl )
∆t
Backward Euler
Approximation
x(tl +1 ) − x(tl )
d
x(tl +1 ) = A x(tl +1 ) ≅
∆t
dt
or
x(tl +1 ) ≅ x(tl ) + ∆t A x(tl +1 )
t
∆ = x(tl +1 ) − ( x(tl ) + ∆t A x(tl +1 ) )
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Finite Difference
Methods
Basic Concepts
Backward Euler Algorithm
Solve with Gaussian Elimination
x
x(t1 ) xˆ1 = x(0) + ∆tAxˆ1
⇒ [ I − ∆tA] xˆ1 = x(0)
x(t2 ) xˆ 2 = [ I − ∆tA] −1 xˆ1
#
x(t L ) xˆ L = [ I − ∆tA] −1 xˆ L −1
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∆tAxˆ 2
∆tAxˆ1
t1
t2
t
Finite Difference
Methods
1 d
d
( x(tl +1 ) + x(tl ))
2 dt
dt
1
= ( Ax(tl +1 ) + Ax(tl ))
2
x(tl +1 ) − x(tl )
∆t
1
x(tl +1 ) x(tl ) + ∆tA( x(tl +1 ) + x(tl ))
2
Basic Concepts
Trapezoidal Rule
slope =
x
d
x(tl )
d
dt
x(tl +1 )
slope =
dt
∆
slope =
x (tl +1 ) − x (tl )
∆t
t
1
1
∆ = ( x(tl +1 ) − ∆tAx(tl )) − ( x(tl ) + ∆tAx(tl +1 ))
2
2
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Basic Concepts
Finite Difference
Methods
Trapezoidal Rule Algorithm
Solve with Gaussian Elimination
∆t
x (t1 ) xˆ = x (0) + ( Ax (0) + Axˆ1 )
2
∆t ⎤ 1 ⎡
∆t ⎤
⎡
⇒ ⎢I −
A⎥ xˆ = ⎢ I +
A⎥ x (0)
2 ⎦
2 ⎦
⎣
⎣
1
∆t
⎡
2
x (t2 ) xˆ = ⎢ I −
2
⎣
#
⎡
x (tL ) xˆ = ⎢ I
⎣
L
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∆t
−
2
−1
∆t
⎤ ⎡
A⎥ ⎢ I +
2
⎦ ⎣
−1
∆t
⎤ ⎡
A⎥ ⎢ I +
2
⎦ ⎣
⎤
A⎥ xˆ1
⎦
⎤
A⎥ xˆ L−1
⎦
x
t1
t2
t
Finite Difference
Methods
Basic Concepts
Numerical Integration View
tl +1
d
x (t ) = A x (t ) ⇒ x (tl +1 ) = x (tl ) + ∫ Ax (τ )dτ
tl
dt
tl +1
∆t
∫tl Ax(τ )dτ ≈ 2 ( Ax(tl ) + Ax(tl ) ) Trap
∆tAx (tl +1 ) BE
∆tAx (tl )
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tl
tl +1
FE
Finite Difference
Methods
Basic Concepts
Summary
Trap Rule, Forward-Euler, Backward-Euler
Are all one-step methods
xˆ l is computed using only xˆ l −1 , not xˆ l − 2 , xˆ l − 3 , etc.
Forward-Euler is simplest
No equation solution
explicit method.
Boxcar approximation to integral
Backward-Euler is more expensive
Equation solution each step
implicit method
Trapezoidal Rule might be more accurate
Equation solution each step
implicit method
Trapezoidal approximation to integral
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Finite Difference
Methods
Numerical Experiments
Unstable Reaction
4
∆t = 0.1
3.5
3
Exact Solution
Backward-Euler
2.5
2
Trap rule
1.5
Forward-Euler
1
0.5
0
0.5
1
1.5
2
FE and BE results have larger errors than Trap Rule,
and the errors grow with time.
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Numerical Experiments
Finite Difference
Methods
Unstable Reaction-Error Plots
0
0.4
6
-0.05
0.35
5
Backward
Euler
0.3
-0.1
x 10
Trap
Rule
4
0.25
3
0.2
2
0.15
1
0.1
0
0.05
-1
-3
-0.15
Forward
Euler
-0.2
-0.25
-0.3
-0.35
0
1
2
0
0
1
2
-2
0
1
All methods have errors which grow exponentially
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2
Numerical Experiments
Finite Difference
Methods
10
M
a
x
E
r
r
o
r
10
10
10
10
10
Unstable Reaction-Convergence
2
Backward-Euler
0
-2
-4
Trap rule
Forward-Euler
-6
-8
10
-3
10
-2
Timestep
10
-1
10
0
For FE and BE, Error ∝ ∆t For Trap, Error ∝ ( ∆t )
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2
Finite Difference
Methods
4
2
Numerical Experiments
Oscillating Strut and Mass
Forward-Euler
∆t = 0.1
0
-2
-4
-6
Trap rule
0
5
10
Backward-Euler
15
20
25
30
Why does FE result grow, BE result decay and the
Trap rule preserve oscillations
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Finite Difference
Methods
Numerical Experiments
Two timescale RC Circuit
small ∆t
Backward-Euler
Computed Solution
large ∆t
With Backward-Euler it is easy to use small timesteps for
the fast dynamics and then switch to large timesteps for the
slow decay
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Finite Difference
Methods
Numerical Experiments
Two timescale RC Circuit
Forward-Euler Computed
Solution
The Forward-Euler is accurate for small timesteps, but goes
unstable when the timestep is enlarged
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Finite Difference
Methods
• Convergence
Numerical Experiments
Summary
– Did the computed solution approach the exact solution?
– Why did the trap rule approach faster than BE or FE?
• Energy Preservation
– Why did BE produce a decaying oscillation?
– Why did FE produce a growing oscillation?
– Why did trap rule maintain oscillation amplitude?
• Two timeconstant (stiff) problems
– Why did FE go unstable when the timestep increased?
We will focus on convergence today
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Finite Difference
Methods
Convergence Analysis
Convergence Definition
Definition: A finite-difference method for solving initial
value problems on [0,T] is said to be convergent if
given any A and any initial condition
max
xˆ − x ( l ∆t ) → 0 as ∆t → 0
l
⎡ T ⎤
l∈⎢0, ⎥
⎣ ∆t ⎦
xˆ l computed with ∆t
∆t
l
xˆ computed with
2
xexact
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Convergence Analysis
Finite Difference
Methods
Order-p convergence
Definition: A finite-difference method for solving initial
value problems on [0,T] is said to be order p convergent
if given any A and any initial condition
max
xˆ − x ( l ∆t ) ≤ C ( ∆t )
l
⎡ T⎤
l∈⎢0, ⎥
⎣ ∆t ⎦
p
for all ∆t less than a given ∆t0
Forward- and Backward-Euler are order 1 convergent
Trapezoidal Rule is order 2 convergent
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Finite Difference
Methods
Convergence Analysis
Two Conditions for
Convergence
1) Local Condition: One step errors are small
(consistency)
Typically verified using Taylor Series
2) Global Condition: The single step errors do not grow
too quickly (stability)
All one-step methods are stable in this sense.
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Finite Difference
Methods
Convergence Analysis
Consistency Definition
Definition: A one-step method for solving initial value
problems on an interval [0,T] is said to be consistent if
for any A and any initial condition
xˆ − x ( ∆t )
1
∆t
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→ 0 as ∆t → 0
Convergence Analysis
Finite Difference
Methods
Consistency for Forward Euler
Forward-Euler definition
1
xˆ = x ( 0 ) + ∆tAx ( 0 )
Expanding in t about zero yields
τ ∈ [0 , ∆ t ]
dx ( 0 ) ( ∆t ) d 2 x (τ )
x(∆t ) = x ( 0 ) + ∆t
+
dt
2
dt 2
2
d
Noting that
x(0) = Ax(0) and subtracting
dt
2
2
Proves the theorem if
( ∆t ) d x (τ ) derivatives of x are
1
xˆ − x ( ∆t ) ≤
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2
dt
2
bounded
Finite Difference
Methods
Convergence Analysis
Convergence Analysis for
Forward Euler
Forward-Euler definition
l +1
l
l
xˆ = xˆ + ∆tAxˆ
Expanding in t about l ∆t yields
x ( ( l + 1) ∆t ) = x ( l ∆t ) + ∆tAx ( l ∆t ) + el
l
where e is the "one-step" error bounded by
e ≤ C ( ∆t ) , where C = 0.5 maxτ ∈[0,T ]
l
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2
d x (τ )
2
dt
2
Finite Difference
Methods
Convergence Analysis
Convergence Analysis for
Forward Euler Continued
Subtracting the previous slide equations
xˆ
l +1
− x ( ( l + 1) ∆t ) = ( I + ∆tA ) ( xˆ − x ( l ∆t ) ) + e
l
l
Define the "Global" error E ≡ x − xˆ ( l ∆t )
l
E
l +1
l
= ( I + ∆tA ) E + e
l
l
Taking norms and using the bound on e
E
l +1
≤ ( I + ∆tA ) E + C ( ∆t )
l
≤ (1 + ∆t A
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)
2
E + C ( ∆t )
l
2
l
Convergence Analysis
Finite Difference
Methods
A helpful bound on difference
equations
A lemma bounding difference equation solutions
If
u
l +1
Then
≤ (1 + ε ) u + b, u = 0, ε > 0
εl
e
l
u ≤
b
ε
l
0
l
To prove, first write u as a power series and sum
l −1
u ≤ ∑ (1 + ε )
l
j =0
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1 − (1 + ε )
b=
b
1 − (1 + ε )
j
j
Convergence Analysis
Finite Difference
Methods
A helpful bound on difference
equations cont.
To finish, note (1 + ε ) ≤ eε ⇒ (1 + ε )l ≤ eε l
εl
1
−
1
+
ε
1
+
ε
−
1
(
)
(
)
e
l
u ≤
b=
b≤
b
1 − (1 + ε )
ε
ε
j
j
Mapping the global error equation to the lemma
E l +1
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⎛
⎞
2
l
⎜
⎟
≤ 1 + ∆t A E + C ( ∆ t )
⎜⎜ N ⎟⎟
ε ⎠
⎝
b
Convergence Analysis
Finite Difference
Methods
Back to Forward Euler
Convergence analysis.
Applying the lemma and cancelling terms
⎛
⎞
l ∆t A
2
e
l
l −1
2
⎜
⎟
E ≤ 1 + ∆t A E + C ( ∆t ) ≤
C
t
∆
(
)
⎜⎜ N ⎟⎟
∆t A
ε ⎠
⎝
b
Finally noting that l ∆t ≤ T ,
max l∈[0, L] E ≤ e
l
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AT
C
∆t
A
Convergence Analysis
Finite Difference
Methods
Observations about the
forward-Euler analysis.
max l∈[0, L] E ≤ e
l
AT
C
∆t
A
• forward-Euler is order 1 convergent
• The bound grows exponentially with time interval
• C is related to the solution second derivative
• The bound grows exponentially fast with norm(A).
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Convergence Analysis
Finite Difference
Methods
Exact and forward-Euler(FE)
Plots for Unstable Reaction.
12
10
RFE
8
Rexact
6
Texact
4
TFE
2
0
0
0.5
1
1.5
2
Forward-Euler Errors appear to grow with time
SMA-HPC ©2003 MIT
2.5
Convergence Analysis
Finite Difference
Methods
1.2
forward-Euler errors for
solving reaction equation.
1
E
0.8
r
r 0.6
o0.4
r
Rexact-RFE
0.2
Texact - TFE
0
-0.2
0
0.5
1
Time
1.5
2
2.5
Note error grows exponentially with time, as bound
predicts
SMA-HPC ©2003 MIT
Convergence Analysis
Finite Difference
Methods
Exact and forward-Euler(FE)
Plots for Circuit.
1
v1exact
0.8
v1FE
0.6
0.4
v2FE
0.2
0
0
v2exact
0.5
1
1.5
2
2.5
3
3.5
4
Forward-Euler Errors don’t always grow with time
SMA-HPC ©2003 MIT
Convergence Analysis
Finite Difference
Methods
forward-Euler errors for
solving circuit equation.
0.03
v1exact - v1FE
E 0.02
r
r 0.01
o
0
r
-0.01
v2exact-v2FE
-0.02
-0.03
0
0.5
1
1.5
Time2
2.5
3
3.5
4
Error does not always grow exponentially with time!
Bound is conservative
SMA-HPC ©2003 MIT
Summary
Initial Value problem examples
Signal propagation (two time scales).
Space frame dynamics (oscillator).
Chemical reaction dynamics (unstable system).
Looked at the simple finite-difference methods
Forward-Euler, Backward-Euler, Trap Rule.
Look at the approximations and algorithms
Experiments generated many questions
Analyzed Convergence for Forward-Euler
Many more questions to answer, some next time