Introduction to Model Order Reduction
Luca Daniel
Massachusetts Institute of Technology
luca@mit.edu
http://onigo.mit.edu/~dluca/2006PisaMOR
www.rle.mit.edu/cpg
1
Course Outline
Numerical Simulation
Quick intro to PDE Solvers
Quick intro to ODE Solvers
Model Order reduction
Linear systems
Common engineering practice
Optimal techniques in terms of model accuracy
Efficient techniques in terms of time and memory
Non-Linear Systems
Parameterized Model Order Reduction
Linear Systems
Non-Linear Systems
2
Outline – ODE solvers:
Numerical Simulation of System Models
• Steady State Analysis
– I.3.a Linear Models
• Gaussian Elimination or LU decomposition
– I.3.b NonLinear Models
• Multidimentional Newton Method
• I.3.c Dynamical Analysis
– Linear Models
• ODE integrators + GE or Krylov
– NonLinear Models
• ODE integrators + Newton
3
Introduction to Model Order Reduction
I.3.a Steady State Analysis
Solving Linear Systems
Luca Daniel
Thanks to Jacob White, Deepak Ramaswamy, Michal
Rewienski, and Karen Veroy
I.3.a Outline
• Linear Systems of Equations
• Gaussian Elimination Cost:
– Bandlimited GE
– Sparse GE
• Krylov Method cost:
– Comparison GE-Krylov
5
Steady State Analysis
dx
 A x(t )  B u (t )
dt
y (t )  C x(t )  D u (t )
• At steady-state u (t )  u
constant,
dx
0
dt
• The analysis boils down to solving a linear system of
equations
b
A x  B u
y C xDu
6
Systems of Linear
Equations


 A1



A2

x1   b1 

    
 x2
b
2 



AN 





     
  xN  bN 



x1 A1  x2 A2    xN AN  b
Find a set of weights, x, so that the weighted
sum of the columns of the matrix M is equal to
the right hand side b
7
Key Questions
Systems of Linear
Equations
• Given Ax = b
– Is there a solution?
– Is the solution Unique?
• Is there a Solution?
There exists weights, x1,
x1 A1  x2 A2 
xN , such that
 xN AN  b
A solution exists when b is in the
span of the columns of M
8
Systems of Linear
Equations
Key Questions
Continued
• Is the Solution Unique?
Suppose there exists weights, y1,
y1 A1  y2 A2 
yN , not all zero
 yN AN  0
Then if Ax  b, therefore A  x  y   b
A solution is unique only if the
columns of A are linearly
independent.
9
Systems of Linear
Equations
Key Questions
Square Matrices
• Given Ax = b, where A is square
– If a solution exists for any b, then the
solution for a specific b is unique.
For a solution to exist for any b, the columns of A must
span all N-length vectors. Since there are only N
columns of the matrix A to span this space, these vectors
must be linearly independent.
A square matrix with linearly independent
columns is said to be nonsingular.
10
I.3.a Outline
• Linear System of Equations
• Gaussian Elimination Cost:
– Bandlimited GE
– Sparse GE
• Krylov Method cost:
– Comparison GE-Krylov
11
Dense GE
or LU decomposition
 A11
A
0
21

A
031

041
A
Picture
A12 A13 A14 

A2222 A
A2323 A2424


A032 A33
A
34
33
34

A
A
A0
A
0
44
43
42
44
43
42
44 
12
Dense GE
Algorithm
For i = 1 to n-1 {
For j = i+1 to n {
“For each Row”
“For each Row below pivot”
A ji
~
A ji 
Aii
Form n-1 reciprocals (pivots)
n 1
n2
Form  (n  i)  multipliers
2
i 1
For k = i+1 to n { “For each element beyond Pivot”
~
~
A jk  A jk  A ji Aik
}
}
}
n 1
2 3
Perform  (n  i)  3 n
i 1
2
Multiply-adds
13
Complexity of dense GE for solving 3-D
FEM or FD metrices
1-D
2-D
3-D
matrix size
n=m
n=m2
n=m3
Domain size = m
1 D
O ( n3 )  O ( m3 )
 100 pt grid O(106 ) ops
2D
O (n3 )  O (m6 )  100 100 grid O(1012 ) ops
3 D
O (n3 )  O (m9 )  100 100 100 grid O(1018 ) ops 317years
10msec
2h 46m
E.g. on a 1 GFlops computer using m=100 and constant c=10
For 2- D and 3-D problems Need a Faster Solver !
14
Triangularizing
Banded GE
Algorithm
b
0
For i = 1 to n-1 {
For j = i+1 to ni+b-1
{ {
For k = i+1 to ni+b-1
{ {
Ajk  Ajk 
NONZEROS
0
Aji
Aii
Aik
}
}
}
n 1
Perform
2
2
(min(
b

1,
n

i
))

O
b
n


i 1
Multiply-adds
15
Complexity of
Banded GE
1-D
2-D
3-D
matrix size
n=m
n=m2
n=m3
band
b=1
b=m
b=m2
Domain size = m
1 D
O (b 2 n)  O (m)
 100 pt grid O(100) ops
2D
O (b 2 n)  O (m4 )  100 100 grid O(108 ) ops
3 D
O (b 2 n)  O (m7 )  100 100 100 grid O(1014 ) ops 11d 13h
1us
1sec
E.g. on a 1 GFlops computer using m=100 and constant c=10
For 3 - D problems Still Need a Faster Solver !
16
Fill-In
Sparse Matrices
Pattern of a Filled-in Matrix

























Very Sparse
Very Sparse
Dense
d

























2-D: d=m
3-D: d=m2
17
Complexity of sparse GE
1-D
2-D
3-D
matrix size
n=m
n=m2
n=m3
dense block
d=1
d=m
d=m2
1-D
O(n) + O(d3) = O(m)
2-D
3-D
Domain size = m
← 100 pt grid
O(100) ops
1us
O(n) + O(d3) = O(m3) ← 100x100 pt grid
O(106) ops
10ms
O(n) + O(d3) = O(m6) ← 100x100x100 pt grid
O(1012) ops 2h 46m
E.g. on a 1 GFlops computer using m=100 and constant c=10
For 3 - D problems Can we do any better
with Krylov subspace iterative methods
18
I.3.a Outline
• Finite Difference Matrices in 1-D, 2-D and 3-D
• Gaussian Elimination Cost:
– Bandlimited GE
– Sparse GE
• Krylov Subspace Iterative cost:
– Comparison GE-Krylov
19
Background
Krylov subspace iterative methods
Problem: solve Ax = b
(where A is large and dense)
guess x0
REPEAT
– how good was my guess? Calculate residue ri = b-Axi
– find next guess xi+1 from the space x0+span{r0,Ar0,A2r0,…,Air0}
which minimizes the next residue
UNTIL ||residue|| < desired accuracy
Advantages over gaussian elimination:
1) get a great control on accuracy (can stop and save
computation when desired accuracy is achieved)
2) only need a matrix vector product Ax per iteration: O(N2)
20
The World According to
Krylov
Work for Banded Gaussian
Elimination, Relaxation and GCR
Dimension Banded GE Sparse GE
1
O  m
2
O m
3

O m 
4
7
GCR GCR+precond

O m 
O m 
O m 
O m 
O  m
O m
2
O(km)
3
3
2
6
4
O(km )
3
O(km )
GCR + good preconditioner ( i.e. k = 5-10 )
is always the best choice for 2D and 3D problems!
21
Steady State Analysis
The World According to
Krylov
Dimension Banded GE Sparse GE
1
2
3
GCR GCR+precond
 
O10us
 m
O10ms
m 
O(0.1ms
km)
4
 
O  m years
317000

O10sec
m 
O10sec
m 
2
7
6
O
m
317 years

4
O2h m
46m
O m
10us
2h
O 46m
m
3
2
3
O0.1sec
(km )
3
20s )
O1m
(km
E.g. on a 1 GFlops computer using m=1000 , k=10 and constant c=10
GCR + good preconditioner ( i.e. k = 5-10 )
is always the best choice for 2D and 3D problems!
22
I.3.a Summary. Steady State
AnalysisLinear Models
• Linear Systems of Equations
• Gaussian Elimination Cost:
– Bandlimited GE
– Sparse GE
• Krylov Method cost:
– Comparison GE-Krylov
23
Outline
Numerical Simulation of System Models
• Steady State Analysis
– I.3.a Linear Models
• Gaussian Elimination or LU decomposition
– I.3.b NonLinear Models
• Multidimentional Newton Method
• I.3.c Dynamical Analysis
– Linear Models
• ODE integrators + GE or Krylov
– NonLinear Models
• ODE integrators + Newton
24
Introduction to Model Order Reduction
I.3.b Steady State Analysis
Solving Non-Linear Systems
Luca Daniel
Thanks to Jacob White, Deepak Ramaswamy,
Jaime Peraire, Michal Rewienski, and Karen Veroy
Steady State
Analysis
( x0 , y0 )
NonLinear Problems
Strut Example
( x1 , y1 )
Given: x0, y0, x1, y1, fl
( x2 , y2 ) 
f l Load force
Find: x2, y2
f

Need to Solve:
f
x
0
y
0
26
Steady State
Analysis
Strut Example
f
( x0 , y0 )
( x1 , y1 )
*
*
fA
fB
( x2 , y2 ) 
f l Load force
 * r *  r
* 
f   *   L0  r  r
r r


r  ( x, y )
*
r  ( x* , y * )

*
A, x
f
*
B, x
 fl , x  0
f A*, y  f B*, y  f l , y  0
f A*, x 
f
*
A, y
y 2  y0

 A LA, 0  LA 
LA
LA 
f B*, x 
f
*
B, y
x2  x0
 A LA, 0  LA 
LA
x2  x0 2   y2  y0 2
x2  x1
 B LB , 0  LB 
LB
y 2  y1

 B LB , 0  LB 
LB
LB 
x2  x1 2   y2  y1 2
27
Steady State
Analysis
Strut Example
Why Nonlinear?
Pull Hard on the Struts
The strut forces change in both
magnitude and direction
Load force

f l  (0,10)
f A*, y  f B*, y  f l , y  0
y 2  y0
y2  y1
 A L0  LA  
 B L0  LB   f l , y  0
LA
LB
28
Steady State
Analysis
F
l x y
2
Example Problem
Strut and Joint
FL
2
(l  lo )
F   Ac
 E (l  lo )
lo
x
x
f x  F  E (l  lo )
l
l
y
y
f y  F  E (l  lo )
l
l
F x 
f x  FLx  0
f y  FLy  0
OR
x
E (l0 lo )  FLx  0
l
y
E (l0 lo )  FLy  0
l
29
Steady State
Analysis
v1
10v
NonLinear Problems
Circuit Example
v2
10
Need to Solve
+
Id  Ir  0
-
I vsrc  I r  0
+- V
d
Branch Equations
1
I r  Vr  0
10
Vd
I d  I s (e
Vt
Non-Linear
System of
Equations
f ( x)  0
 1)  0
30
Steady State
Analysis
v1
i2
i1
Example Problem
Nonlinear Resistors
v2
i3
Nodal Analysis
At Node 1: i1  i2  0
 g  v1   g  v1  v2   0
At Node 2: i3  i2  0
Nonlinear
Resistors
i  g v
 gg(v
 v23)   g  v1  v2   0
Two coupled
nonlinear equations
in two unknowns
31 1:33
Steady State Analysis
dx
 F ( x(t ) , u (t ) )
dt
y (t )  G ( x(t ) , u (t ) )
• At steady-state u (t )  u
constant,
dx
0
dt
• The analysis boils down to solving a non-linear
system of equations
F ( x, u )  0
y  G ( x, u )
32
Newton Idea
1-D Reminder
 
Problem: Find x such that f x  0
*
*
Approximate f(x) with its Taylor series about xk:
a straight line
ˆf ( x )  f ( x k )  f ( x k ) ( x  x k )
x
Find the solution of the approximation: xk+1
1
f k k 1 k
 f k 
k
k 1
k
f ( x )  ( x ) ( x  x )  0  x  x   ( x ) f ( x k )
x
 x

33
1-D Reminder
Newton Algorithm
Algorithm Picture
34 1:20
Newton Algorithm
1-D Reminder
x = Initial Guess, k  0
0
Repeat {
1
 f k 
x  x   ( x ) f ( x k )
 x

k  k 1
k 1
k
} Until ?
x
k 1
 x  threshold ?
k

f x
k 1
  threshold ?
35
Newton’s Method
Convergence
Example
Convergence Depends on a Good Initial Guess
f(x)
1
x
1
x
x
2
x
0
x
0
X
36 1:27
Multidimensional
Newton Method
Jacobian Matrix
J F  x  x  F  x  x   F  x 
 F1  x 

 x1
J F  x  x  

 FN  x 
 x
1

F1  x  

xN   x1 




FN  x    xN 

xN 
37 1:35
Multidimensional
Newton Method
General Setting
 
Problem: Find x such that F x  0
*
x R
*
N
*
and F : R  R
N
N
Approximate F(x) with its Taylor series about xk:
k
k
k
ˆ
F ( x)  F ( x )  J F ( x ) ( x  x )
Jacobian
Find the solution of the approximation: xk+1
F(x )  JF (x ) (x
k

k
x
k 1

k 1
x )0
k
 x  JF (x )
k
k

1
k
F(x )
38
Multidimensional
Newton Method
Newton Algorithm
x = Initial Guess, k  0
0
Repeat {
   
 x  x  x    F  x  for x
Compute F x k , J F x k
Solve J F
k
k 1
k
k
k 1
k  k 1
} Until
x k 1  x k ,

f x k 1

small enough
39 1:39
Multidimensional
Newton Method
Nodal Analysis
Strut and Joint
x R and F : R  R
*
2
2
F
x
E (l0  lo )  FLx  0
l
y
E (l0  lo )  FLy  0
l
2
FL
? ? 
JF  x  

?
?


40
Multidimensional
Newton Method
i1
Nonlinear Resistor
x R and F : R  R
*
v1
Nodal Analysis
i2
B
v2
i3
2
2
2
At Node 1: i1  i2  0
 F1  v   g  v1   g  v1  v2   0
A
B
At Node 2: i3  i2  0
 F2  v   gC(v
v32)  g  v1  v2   0
B
? ? 
JF  x  

? ? 
41 1:43
Computing the Jacobian
and the Function
Multidimensional
Newton Method
Consider the contribution of one nonlinear
resistor Connected between nodes n1 and n2
i b + vb -
n1
 
i g v
b
n2
b


Summing currents at Node n1: Fn1  v   g vn1  vn2 


Summing currents at Node n 2 : Fn2  v    g vn1  vn2 
Differenting at Node n1:
Fn1  v 
vn1


g vn1  vn2
vn1
g
v

Fn1  v 
vn2


g vn1  vn2
vn2


g
v
42
Computing the Jacobian
and the Function
Multidimensional
Newton Method
i
Stamping a
Resistor
n1
n2














v


g vn1  vn2
+
-
n1
n2
n1
g vn1  vn2
vb


g vn1  vn2
v

g vn1  vn2
v
v
JF v 
n2















 g vn1  vn2


 g v  v
n1
n2


F (v )






 n1


n
 2

43 1:47
More Complete
Multidimensional
Newton Algorithm
Newton Method
0
x = Initial Guess, k  0
Repeat {
 
 
k
k
Compute
F
x
,
J
x
F
Zero J F and F
for each element
Compute element currents and derivatives
Sum currents to F , sum derivatives to J F
 
Solve J F x k

 
x k 1  x k   F x k for x k 1
k  k 1
} Until
x k 1  x k ,

f x k 1

small enough
44
Multidimensional
Newton’s Method
Example: Heat Flow
in immersed bar
T 0
T 0
T1
TN
ih  kT
3
What is the Jacobian?
45 1:53
Outline
Numerical Simulation of System Models
• Steady State Analysis
– I.3.a Linear Models
• Gaussian Elimination or LU decomposition
– I.3.b NonLinear Models
• Multidimentional Newton Method
• I.3.c Dynamical Analysis
– Linear Models
• ODE integrators + GE or Krylov
– NonLinear Models
• ODE integrators + Newton
46
Introduction to Model Order Reduction
I.3.c - Dynamical Analysis of Systems
ODE numerical integrators
Luca Daniel
Thanks to Jacob White, Deepak Ramaswamy, Jaime
Peraire, Michal Rewienski, and Karen Veroy
47
Dynamical Analysis of Systems
dx
 A x(t )  B u (t )
dt
y (t )  C x(t )  D u (t )
dx
 F ( x(t ) , u (t ) )
dt
y (t )  G ( x(t ) , u (t ) )
• Given a generic input u(t) calculate x(t) and y(t)
• The analysis boils down to integrating numerically a
system of linear or non-linear ordinary differential
equations
48
Basic Concepts
Finite Difference
Methods
t t
First - Discretize Time
0 t1 t2
Second - Represent x(t) using values at ti
3
x̂ 2
4
x̂
1
x̂
x̂
tL
0 t1 t2 t3
dx
Third - approximat e
dt
using the discrete
?

d
xˆ l 1  xˆ l
Example :
x(tl ) 
dt
tl 1
?

d
x(tl 1 )
dt
t
t L 1 tL  T
x(tl )

Approx.
Exact
xˆ
l
solution
solution
xˆ l
49 1:55
Finite Difference
Methods
x
d
slope  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 ) 
dt
t
or
x(tl 1 )  x(tl )  t A x(tl )
t
  x(tl 1 )   x(tl )  t A x(tl ) 
50
Finite Difference
Methods
Basic Concepts
Forward Euler Algorithm
x(t1 )  xˆ1  x(0)  tAx  0 
x
x(t2 )  xˆ 2  xˆ1  tAxˆ1
tAxˆ1
x(t L )  xˆ L  xˆ L 1  tAxˆ L 1
tAx(0)
t1
t2
t3
t
51 1:59
Basic Concepts
Finite Difference
Methods
x
d
slope  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 ) 
dt
t
or
x(tl 1 )  x(tl )  t A x(tl 1 )
t
  x(tl 1 )   x(tl )  t A x(tl 1 ) 
52 2:01
Finite Difference
Methods
Basic Concepts
Backward Euler Algorithm
Solve with GE or iterative methods
x(t1 )  xˆ1  x(0)  tAxˆ1
 [ I  tA] xˆ1  x(0)
x(t2 )  xˆ 2  [ I  tA] 1 xˆ1
x(tL )  xˆ L  [ I  tA] 1 xˆ L 1
x
tAxˆ 2
tAxˆ1
t1
t2
t
53 2:03
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 )
dt
slope 

slope 
d
x(tl 1 )
dt
x (tl 1 )  x (tl )
t
t
1
1
  ( x(tl 1 )  tAx(tl ))  ( x(tl )  tAx(tl 1 ))
2
2
54 2:05
Finite Difference
Methods
Basic Concepts
Trapezoidal Rule Algorithm
Solve with GE or iterative methods
t
x (t1 )  xˆ  x (0)   Ax(0)  Axˆ1 
2
t  1 
t 

  I  A xˆ   I  A x (0)
2 
2 


1
t

x (t2 )  xˆ 2   I 
2


x (tL )  xˆ L   I

t

2
1
t
 
A  I 
2
 
1
t
 
A  I 
2
 

A xˆ1

x
t1
t2
t

A xˆ L1

55
Finite Difference
Methods
Basic Concepts
Summary
Trap Rule, Forward-Euler, Backward-Euler
Are all one-step methods
ˆx 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.
Backward-Euler is more expensive
Equation solution each step
implicit method
Trapezoidal Rule might be more accurate
Equation solution each step
implicit method
56 2:13
Outline
• Initial Value problem examples
Signal propagation (circuits with capacitors).
Space frame dynamics (struts and masses).
Chemical reaction dynamics.
• Eigenvalue analysis for dynamical systems
• Investigate the simple finite-difference methods
Forward-Euler, Backward-Euler, Trap Rule.
Examine properties experimentally.
57 2:15
Application
Problems
Chemical Reaction
Dynamics
Crucible
Reagent
Strange green
stuff
• How fast is product produced?
• Does it explode?
58
Application
Problems
Chemical Reaction
Example
A 2x2 Example
Amount of reactant  R, the temperature  T
dT
 T  R
dt
dR
  R  4T
dt
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.
59
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.
60 2:15
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
2
All methods have errors which grow exponentially
61 2:17
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 
2
62 2:19
Finite Difference
Methods
Basic Concepts
Numerical Integration View
tl 1
d
x(t )  A x (t )  x (tl 1 )  x (tl )   Ax( )d
tl
dt
tl 1
t
Trap
Ax
(
t
)

Ax
(
t
)
Ax
(

)
d




l
l
tl
2
tAx(tl 1 ) BE
tAx(tl )
tl
tl 1
FE
63 2:09
Application
Problems
y  y0  u
Struts, Joints and point
mass example
A 2x2 Example
Constitutive
Equations
fs
fm
y  y0 EAc
f s  E Ac 

u
y0
y0
Conservation
Law
fs  fm  0
d 2u
fm  M 2
dt
Define v as velocity (du/dt) to yield a 2x2 System
M
0

 dv  
EAc 
0   dt  0 
v 

y0  
 

 u 
1   du 
0 
 dt  1
64 1:39
Finite Difference
Methods
4
2
Numerical Experiments
Oscillating Strut and Mass
t  0.1
Forward-Euler
0
-2
-4
-6
Trap rule
0
5
10
Backward-Euler
15
20
25
30
FE result grow, BE result decay and the Trap rule
preserves oscillations
65
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
66
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
67
NonLinear Systems
ODE integrators
Multistep Methods
Nonlinear Differential Equation:
k-Step Multistep Approach:
l 2
xˆ
l k
xˆ ˆ l 1 l
x xˆ
d
x(t )  f ( x(t ), u (t ))
dt

l j
l j
ˆ
ˆ

x


t

f
x
, u  tl  j 
 j
 j
k
k
j 0
j 0
Multistep coefficients
Solution at discrete points
tl  k
tl 3 tl  2 tl 1 tl
Time discretization
68

Basic Equations
Multistep Methods
Common Algorithms
Multistep Equation:
Forward-Euler Approximation:
FE Discrete Equation:
Multistep Coefficients:
BE Discrete Equation:
Multistep Coefficients:

l j
l j
ˆ
ˆ

x


t

f
x
, u  tl  j 
 j
 j
k
k
j 0
j 0

x  tl   x  tl 1   t f  x tl 1  , u tl 1  
xˆ l  xˆ l 1  t f  xˆ l 1 , u  tl 1  
k  1, 0  1, 1  1, 0  0, 1  1
xˆ l  xˆ l 1  t f  xˆ l , u  tl  
k  1, 0  1, 1  1, 0  1, 1  0
t
f  xˆ l , u  tl    f  xˆ l 1 , u  tl 1  
Trap Discrete Equation:
2
1
1
Multistep Coefficients: k  1,  0  1, 1  1,  0  , 1 
2
2
xˆ l  xˆ l 1 


69
Outline
Numerical Simulation of System Models
• Steady State Analysis
– I.3.a Linear Models
• Gaussian Elimination or LU decomposition
– I.3.b NonLinear Models
• Multidimentional Newton Method
• I.3.c Dynamical Analysis
– Linear Models
• ODE integrators + GE or Krylov
– NonLinear Models
• ODE integrators + Newton
70