CSE245: Computer-Aided Circuit Simulation and

VerificationLecture Note 5

Numerical Integration

Prof. Chung-Kuan Cheng

1

Numerical Integration: Outline

• One-step Method for ODE (IVP)– Forward Euler– Backward Euler– Trapezoidal Rule– Equivalent Circuit Model

• Convergence Analysis• Linear Multi-Step Method• Time Step Control

2

Ordinary Difference Equations

.condition initial given the intervalan in

)(

),()(

:(IVP) Problem Value Initial Solve

00

00

x,T][t

xtx

txfdt

tdx

N equations, n x variables, n dx/dt.

Typically analytic solutions are not available

solve it numerically

3

Numerical Integration

Forward Euler

Backward Euler

Trapezoidal

0 0

( )( , )

( )

dx tf x t

dtx t x

4

Numerical Integration: State Equation

Forward Euler

Backward Euler

5

Numerical Integration: State Equation

Trapezoidal

6

7

Equivalent Circuit Model-BE• Capacitor

( ) ( ) ( )tCv t t v t i t t

+

C

-

+

-

( )v t t

( )i t t

( )i t t

Ceq tG

( )v t t

+

-( )C

eq tI v t

( ) ( ) ( )C Ct ti t t v t t v t

8

Equivalent Circuit Model-BE• Inductor

( ) ( ) ( )tLi t t i t v t t

+

L

-

+

-

( )v t t

( )i t t

( )i t t

Leq tR

( )v t t

+

-( )L

eq tV i t

( ) ( ) ( )L Lt tv t t i t t i t

9

Equivalent Circuit Model-TR• Capacitor

2( ) ( ) ( ( ) ( ))tCv t t v t i t i t t

+

C

-

+

-

( )v t t

( )i t t

( )i t t

2Ceq tG

( )v t t

+

- 2 ( ) ( )Ceq tI v t i t

2 2( ) ( ) ( ) ( )C Ct ti t t v t t v t i t

10

Equivalent Circuit Model-TR• Inductor

2( ) ( ) ( ( ) ( ))tLi t t i t v t v t t

+

L

-

+

-

( )v t t

( )i t t

( )i t t

2Leq tR

( )v t t

+

-2 ( ) ( )L

eq tV i t v t

2 2( ) ( ) ( ) ( )L Lt tv t t i t t i t v t

Trap Rule, Forward-Euler, Backward-Euler All are one-step methods xk+1 is computed using only xk, not xk-1, xk-2, xk-3... Forward-Euler is the simplest No equation solution explicit method. Backward-Euler is more expensive Equation solution each step implicit method most stable (FE/BE/TR) Trapezoidal Rule might be more accurate Equation solution each step implicit method More accurate but less stable, may cause oscillation

Summary of Basic Concepts

11

12

Stabilities

Froward Euler

0 -1

h

stable

unstable

j

1k k k

k k

x x hx

x x

1k k kx x h x

11 0(1 ) (1 )kk kx h x h x

Difference EqnStability region 1-1

1z h

Im(z)

Re(z)

Im

Re

Forward Euler

ODE stability region

2

t

Region ofAbsolute Stability

FE region of absolute stability

13

14

Stabilities

Backward Euler

1 1

1 1

k k k

k k

x x hx

x x

1 1k k kx x h x

11 0

1 1( )

1 1k

k kx x xh h

0 1

h

1-h

stable

unstable

j

Difference EqnStability region 1-1

Im(z)

Re(z)

Im Backward Euler 1

1z h

Region ofAbsolute Stability

BE region of absolute stability

15

16

Stabilities

Trapezoidal

1 1

1 1

( )2k k k k

k k

k k

hx x x x

x x

x x

1 1( )2k k k k

hx x x x

11 0

1 12 2( )

1 12 2

kk k

h h

x x xh h

0 1

h

1+h/2

stable

unstable

-1

1-h/2

j

Convergence• Consistency: A method of order p (p>1) is

consistent if

• Stability: A method is stable if:

• Convergence: A method is convergent if:

Consistency + Stability Convergence

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A-Stable

• Dahlquist Theorem:– An A-Stable LMS (Linear Multi-Step) method

cannot exceed 2nd order accuracy• The most accurate A-Stable method

(smallest truncation error) is trapezoidal method.

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Convergence Analysis: Truncation Error

• Local Truncation Error (LTE):– At time point tk+1 assume xk is exact, the difference between

the approximated solution xk+1 and exact solution x*k+1 is called

local truncation error.– Indicates consistency– Used to estimate next time step size in SPICE

• Global Truncation Error (GTE):– At time point tk+1, assume only the initial condition x0 at time t0

is correct, the difference between the approximated solution xk+1 and the exact solution x*

k+1 is called global truncation error.

– Indicates stability

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LTE Estimation: SPICE• Taylor Expansion of xn+1 about the time point tn:

x(tn+1)=x(tn)+dx(tn)/dt∙h+d2x(tn)/dt2∙h2/2!+d3x(tn)/dt3∙h3/3!+…

• Taylor Expansion of xn about the time point tn+1: x(tn)=x(tn+1)-dx(tn+1)/dt∙h+d2x(tn+1)/dt2∙h2/2!-d3x(tn+1)/dt3∙h3/3!+…

• Forward Euler– Exercise x(tn+1)-xn+1=x(tn+1)-(xn+hdxn/dt)

• Backward Euler– Exercise x(tn+1)-xn+1=x(tn+1)-(xn+hdxn+1/dt)

• Trapezoidal– Exercise x(tn+1)-xn+1=x(tn+1)-[xn+(dxn/dt+dxn+1/dt)h/2]

LTE 20

Formula for pth order method

LTE 21

Formula E(x(t),h)=∑i=0,kaix(tn-i)+hbidx(tn-i)/dt

Let x(t)=[(tn-t)/h]p

E(x(t),h)=∑i=0,kai[(tn-tn-i)/h]p-pbi[(tn-tn-i)/h]p-1

=∑i=0,kaiip-pbiip-1

If the formula is a pth order method, we have

Case p=0: ∑i=0,kai=0

Case p=1: ∑i=0,kaii-bi=0

……

Case p: ∑i=0,k(aii-pbi)ip-1=0

Formula for pth order method: Example

LTE 22

Forward Euler: We have a0=1, a1=-1, b0=0, b1=-1

Case p=0: ∑i=0,kai=0

Case p=1: ∑i=0,kaii-bi=0

Case p=2: ∑i=0,k(aii-2bi)i=-1+2=1

Backward Euler: We have a0=1, a1=-1, b0=-1, b1=0

Case p=0: ∑i=0,kai=0

Case p=1: ∑i=0,kaii-bi=0

Case p=2: ∑i=0,k(aii-2bi)i=-1+2=1

Trapezoidal Rule: We have a0=1, a1=-1, b0=-1/2, b1=-1/2

Case p=0: ∑i=0,kai=0

Case p=1: ∑i=0,kaii-bi=0

Case p=2: ∑i=0,k(aii-2bi)i=0

Case p=3: ∑i=0,k(aii-3bi)i2=-1+3/2=1/2

Formula for pth order method: Variables

LTE 23

There are 2(k+1)-1 unknowns (a0=1), and p+1 equations.

Thus, we need 2(k+1)-1 ≥ p+1

In other words, k ≥ p/2

Formula for pth order method: Local Truncation Error

LTE 24

By Taylor’s expansion, we have

x(t)=x(tn)+x’(tn)(t-tn)+…+1/(p+1)! x(p+1)(tn)(t-tn)p+1+…

Thus, the error of pth order method is

∑i=0,k{ai[(tn-tn-i)/h]p+1-pbi[(tn-tn-i)/h]p}1/(p+1)! x(p+1)(tn)hp+1+O(hp+2)

Let us set Ep+1={∑i=0,kaiip+1-pbiip}/p!

Method a0 a1 b0 b1 Ep+1

FE 1 -1 0 -1 E2=1/2

BE 1 -1 -1 0 E2=1/2

TR 1 -1 -1/2 -1/2 E3=1/12

Time Step Control: SPICE• We have derived the local truncation error

the unit is charge for capacitor and flux for inductor• Similarly, we can derive the local truncation error in

terms of (1)

the unit is current for capacitor and voltage for inductor• Suppose ED represents the absolute value of error that is

allowed per time point. That is together with (1) we can calculate the time step as

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Time Step Control: SPICE (cont’d)• DD3(tn+1) is called 3rd divided difference, which is given

by the recursive formula

26

Multiple Step Integration: Stability

For a system x’=λx, let q= λh.

The integration formula is ∑i=0,kaixn-i+hbix’n-i=0.

We set (a0+qb0)zk+ (a1+qb1)zk-1 …+(ak+qbk)=0

There are k roots, ri, of the polynomial eq.

The generic solution is xn=c1r1n+c2r2

n+…+ckrkn

but for multiplicity root, we have xn=…+(ci0+ci1n+…+cimnm-1)ri

n+…

If |ri|< 1 for all i, the system is stable

Else for |ri|= 1 but not multiplicity root, the system remains stable

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Multiple Step Integration: Stability

For a system x’=λx, let q= λh.

The integration formula is ∑i=0,kaixn-i+hbix’n-i=0.

We set (a0+qb0)zk+ (a1+qb1)zk-1 …+(ak+qbk)=0

Examples using FE, BE, TZ methods

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Method a0 a1 b0 b1 root

FE 1 -1 0 -1 1+q

BE 1 -1 -1 0 1/(1-q)

TR 1 -1 -1/2 -1/2 (1+q/2)/(1-q/2)

Multiple Step Integration: Stability

For a system x’=λx, let q= λh.

The integration formula is ∑i=0,kaixn-i+hbixn-i=0.

A Stability: The system is stable for all Real(q)≤ 0.

Dahlquist’s barrier:– An A-Stable LMS (Linear Multi-Step) method

cannot exceed 2nd order accuracy

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