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Methods for Ordinary Differential Equations. Lecture 10 Alessandra Nardi. Thanks to Prof. Jacob White, Deepak Ramaswamy Jaime Peraire, Michal Rewienski, and Karen Veroy. Outline. Transient Analysis of dynamical circuits i.e., circuits containing C and/or L Examples - PowerPoint PPT Presentation
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Methods for Ordinary Differential Equations
Lecture 10
Alessandra Nardi
Thanks to Prof. Jacob White, Deepak Ramaswamy Jaime Peraire, Michal Rewienski, and Karen Veroy
Outline
• Transient Analysis of dynamical circuits– i.e., circuits containing C and/or L
• Examples
• Solution of Ordinary Differential Equations (Initial Value Problems – IVP)– Forward Euler (FE), Backward Euler (BE) and
Trapezoidal Rule (TR)– Multistep methods– Convergence
Ground Plane
Signal Wire
LogicGate
LogicGate
• Metal Wires carry signals from gate to gate.• How long is the signal delayed?
Wire and ground plane form a capacitor
Wire has resistance
Application ProblemsSignal Transmission in an Integrated Circuit
capacitor
resistor
• Model wire resistance with resistors.• Model wire-plane capacitance with capacitors.
Constructing the Model• Cut the wire into sections.
Application ProblemsSignal Transmission in an IC – Circuit Model
Nodal Equations Yields 2x2 System
C1
R2
R1 R3 C2
Constitutive Equations
cc
dvi C
dt
1R Ri v
R
Conservation Laws
1 1 20C R Ri i i
2 3 20C R Ri i i
1
1 2 21 1
2 22
2 3 2
1 1 1
0
0 1 1 1
dvR R RC vdt
C vdv
R R Rdt
1Ri
1Ci
2Ri
2Ci
3Ri
1v 2v
Application ProblemsSignal Transmission in an IC – 2x2 example
eigenvectors
1
1 2 21 1
2 22
2 3 2
1 1 1
0
0 1 1 1
dvR R RC vdt
C vdv
R R Rdt
1 2 1 3 2Let 1, 10, 1C C R R R 1.1 1.0
1.0 1.1
A
dxx
dt
11 1 0.1 0 1 1
1 1 0 2.1 1 1A
Eigenvalues and Eigenvectors
Eigenvalues
Application ProblemsSignal Transmission in an IC – 2x2 example
1Change of variab (le ) ( ) ( ) (s ): Ey t x t y t E x t
1
1 2 1 2
10 0
0 0
0 0n n
n
A E E E E E
E
E
Eigendecomposition:
0
( )Substituting: ( ), (0)
dEy tAEy t Ey x
dt
11Multiply by ( ) : ( )dy t
E AE tdt
E y 0 0
10 0
0 0
( )n
y t
0Consider an ODE( )
( ), (0): dx t
Ax t x xdt
An Aside on Eigenanalysis
( )Decoupling: ( ) ( ) (0)iti
i i i
dy ty t y t e y
dt
1 0 0
0 0
0 0
From last slide:( ) ( )
n
dy tdt
y t
DecoupledEquations!
1) Determine , E 1 0 0
3) Compute ( ) 0 0 (0)
0 0 n
t
t
e
y t y
e
102) Compute (0) y E x
4) ( ) ( ) x t Ey t
0Steps for solvi( )
( ), (0)ng dx t
Ax t x xdt
An Aside on Eigenanalysis
1(0) 1v
2 (0) 0v
Notice two time scale behavior• v1 and v2 come together quickly (fast eigenmode).• v1 and v2 decay to zero slowly (slow eigenmode).
Application ProblemsSignal Transmission in an IC – 2x2 example
Circuit Equation Formulation
• For dynamical circuits the Sparse Tableau equations can be written compactly:
• For sake of simplicity, we shall discuss first order ODEs in the form:
riablescircuit va of vector theis where
)0(
0),,)(
(
0
x
xx
txdt
tdxF
),()(
txfdt
tdx
Ordinary Differential EquationsInitial Value Problems (IVP)
Typically analytic solutions are not available
solve it numerically
.condition initial given the intervalan in
)(
),()(
:(IVP) Problem Value Initial Solve
00
00
x,T][t
xtx
txfdt
tdx
• Not necessarily a solution exists and is unique for:
• It turns out that, under rather mild conditions on the continuity and differentiability of F, it can be proven that there exists a unique solution.
• Also, for sake of simplicity only consider
linear case:
0),,( tydt
dyF
We shall assume that has a unique solution 0),,( tydt
dyF
00 )(
)()(
xtx
tAxdt
tdx
Ordinary Differential Equations Assumptions and Simplifications
First - Discretize Time
Second - Represent x(t) using values at ti
ˆ ( )llx x t
Approx. sol’n
Exact sol’n
Third - Approximate using the discrete ( )ldx t
dtˆ 'slx
1 1
1
ˆ ˆ ˆ ˆExample: ( )
l l l l
ll l
d x x x xx t or
dt t t
Lt T1t 2t 1Lt 0
t t t
1t 2t 3t Lt0
3x̂ 4x̂1x̂
2x̂
Finite Difference MethodsBasic Concepts
lt 1lt t
x
1( ) ( )slope l lx t x t
t
slope ( )ldx t
dt
1( ) ( ) ( )l l lx t x t t A x t
1
1
( ) ( )( ) ( )
or
( ) ( ) ( )
l ll l
l l l
x t x tdx t A x t
dt t
x t x t t A x t
Finite Difference Methods Forward Euler Approximation
1t 2t t
x
(0)tAx
11 ˆ( ) (0) 0x t x x tAx
3t
2 1 12 ˆ ˆ ˆ( )x t x x tAx
1ˆtAx
1 1ˆ ˆ ˆ( ) L L LLx t x x tAx
Finite Difference Methods Forward Euler Algorithm
lt 1lt t
x 1slope ( )l
dx t
dt
1( ) ( )slope l lx t x t
t
1 1( ) ( ) ( )l l lx t x t t A x t
11 1
1 1
( ) ( )( ) ( )
or
( ) ( ) ( )
l ll l
l l l
x t x tdx t A x t
dt t
x t x t t A x t
Finite Difference Methods Backward Euler Approximation
1t 2t t
x
1ˆtAx
2ˆtAx
1 11 ˆ ˆ( ) (0)x t x x tAx
Solve with Gaussian Elimination
1ˆ[ ] (0)I tA x x
1 1ˆ ˆ( ) [ ]L LLx t x I tA x
2 1 12 ˆ ˆ( ) [ ]x t x I tA x
Finite Difference Methods Backward Euler Algorithm
1
1
1
1 1
1( ( ) ( ))
21
( ( ) ( ))2
( ) ( )
1( ) ( ) ( ( ) ( ))
2
l l
l l
l l
l l l l
d dx t x t
dt dt
Ax t Ax t
x t x t
t
x t x t tA x t x t
t
x
1( ) ( )slope l lx t x t
t
slope ( )ldx t
dt
1slope ( )l
dx t
dt
1 1
1 1( ( ) ( )) ( ( ) ( ))
2 2l l l lx t tAx t x t tAx t
Finite Difference Methods Trapezoidal Rule Approximation
1t 2t t
x
1ˆ (0)2 2
t tI A x I A x
Solve with Gaussian Elimination
1 11 ˆ ˆ( ) (0) (0)
2
tx t x x Ax Ax
12 1
2
11
ˆ ˆ( )2 2
ˆ ˆ( )2 2
L LL
t tx t x I A I A x
t tx t x I A I A x
Finite Difference Methods Trapezoidal Rule Algorithm
1
1( ) (( ) )) )( (l
l
t
l l t
dx t x t x t A dx t xA
dt
lt 1lt
1
( )l
l
t
tAx d
1( )ltAx t BE
( )ltAx t FE
( ) ( )2 l l
tAx t Ax t
Trap
Finite Difference Methods Numerical Integration View
Trap Rule, Forward-Euler, Backward-Euler Are all one-step methods
Forward-Euler is simplest No equation solution explicit method. Boxcar approximation to integralBackward-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
1 2 3ˆ ˆ ˆ ˆ is computed using only , not , , etc.l l l lx x x x
Finite Difference Methods Summary of Basic Concepts
( ) ( ( ), ( ))dx t f x t u t
dtNonlinear Differential Equation:
k-Step Multistep Approach: 0 0
ˆ ˆ ,k k
l j l jj j l j
j j
x t f x u t
Solution at discrete points
Time discretization
Multistep coefficients
2ˆ lx
lt1lt 2lt 3lt l kt
ˆ lx1ˆ lx
ˆ l kx
Multistep Methods Basic Equations
Multistep Equation:
1 1 1,l l l lx t x t t f x t u t
FE Discrete Equation: 1 11ˆ ˆ ˆ ,l l l
lx x t f x u t
0 1 0 11, 1, 1, 0, 1k
Forward-Euler Approximation:
Multistep Coefficients:
Multistep Coefficients:
BE Discrete Equation:
0 1 0 11, 1, 1, 1, 0k 1ˆ ˆ ˆ ,l l l
lx x t f x u t
Trap Discrete Equation: 1 11ˆ ˆ ˆ ˆ, ,
2l l l l
l l
tx x f x u t f x u t
0 1 0 1
1 11, 1, 1, ,
2 2k Multistep Coefficients:
0 0
ˆ ˆ ,k k
l j l jj j l j
j j
x t f x u t
Multistep Methods – Common AlgorithmsTR, BE, FE are one-step methods
Multistep Equation:
01) If 0 the multistep method is implicit 2) A step multistep method uses previous ' and 'k k x s f s
03) A normalization is needed, 1 is common
4) A -step method has 2 1 free coefficientsk k
How does one pick good coefficients?
Want the highest accuracy
0 0
ˆ ˆ ,k k
l j l jj j l j
j j
x t f x u t
Multistep Methods Definition and Observations
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
0,
ˆmax 0 as t 0lT
lt
x x l t
exactx
tˆ computed with
2lx
ˆ computed with tlx
Multistep Methods – Convergence Analysis
Convergence Definition
Definition: A multi-step method for solving initial value problems on [0,T] is said to be order p
convergent if given any A and any initial condition
0,
ˆmaxpl
Tl
t
x x l t C t
0for all less than a given t t
Forward- and Backward-Euler are order 1 convergentTrapezoidal Rule is order 2 convergent
Multistep Methods – Convergence Analysis
Order-p Convergence
Multistep Methods – Convergence Analysis
Two types of error
made.been haserror previous no assuming ,solution theof value
exact theand ˆ valuecomputed ebetween th difference theis
at methodn integratioan of (LTE)Error Truncation Local The
1
1
1
)x(t
x
t
l
l
l
exactly.known iscondition
initial only the that assuming ,solution theof value
exact theand ˆ valuecomputed ebetween th difference theis
at methodn integratioan of (GTE)Error Truncation Global The
1
1
1
)x(t
x
t
l
l
l
• For convergence we need to look at max error over the whole time interval [0,T]– We look at GTE
• Not enough to look at LTE, in fact:– As I take smaller and smaller timesteps t, I would
like my solution to approach exact solution better and better over the whole time interval, even though I have to add up LTE from more timesteps.
Multistep Methods – Convergence Analysis
Two conditions for Convergence
1) Local Condition: One step errors are small (consistency)
2) Global Condition: The single step errors do not grow too quickly (stability)
Typically verified using Taylor Series
All one-step methods are stable in this sense.
Multistep Methods – Convergence Analysis
Two conditions for Convergence
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
1ˆ0 as t 0
x x t
t
One-step Methods – Convergence Analysis
Consistency definition
Forward-Euler definition
22
2
0()
2 0
dxtdxxtxt
dtdt
Expanding in about zero yields t
1ˆ00 xxtAx
Noting that (0)(0) and subtractingdxAx
dt
22
12 ˆ
2
tdxxxt
dt
Proves the theorem if
derivatives of x are bounded
0,t
One-step Methods – Convergence Analysis
Consistency for Forward Euler
Forward-Euler definition
1l
xltxlttAxlte
Expanding in about yields tlt 1
ˆˆ ˆlllxxtAx
where is the "one-step" error bounded byle
2
2
[0,]2 , where 0.5maxl
T
dxeCtC
dt
One-step Methods – Convergence Analysis
Convergence Analysis for Forward Euler
ˆ Define the "Global" errorll
Exxlt
1ˆˆ 1lllxxltItAxxlte
Subtracting the previous slide equations
Taking norms and using the bound on le
1 lll
EItAEe
2 1
21
ll
l
EItAECt
tAECt
One-step Methods – Convergence Analysis
Convergence Analysis for Forward Euler
10
1,0, If 0ll
uubu
To prove, first write as a power series and sum l
u
1
0
111
11
l lj l
j
ubb
Then l
leub
A lemma bounding difference equation solutions
One-step Methods – Convergence Analysis
A helpful bound on difference equations
To finish, note (1)(1) ll
ee
1111
11
lllle
ubbb
2 1
1ll
EtAECt
b
Mapping the global error equation to the lemma
One-step Methods – Convergence Analysis
A helpful bound on difference equations
2 1
1ll
EtAECt
b
Applying the lemma and cancelling terms
2
ltAe
CttA
Finally noting that , ltT
0, maxAT l
lL
CEet
A
One-step Methods – Convergence Analysis
Back to Convergence Analysis for Forward Euler
0, maxAT l
lL
CEet
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).
One-step Methods – Convergence Analysis
Observations about Convergence Analysis for FE
Summary
• Transient Analysis of dynamical circuits– i.e., circuits containing C and/or L
• Examples
• Solution of Ordinary Differential Equations (Initial Value Problems – IVP)– Forward Euler (FE), Backward Euler (BE) and
Trapezoidal Rule (TR)– Multistep methods– Convergence