# General Linear Methods for Integrated Circuit Design

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## General Linear Methodsfor Integrated Circuit Design

Steffen Voigtmann

Oberwolfach, April 2006

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Integrated circuit design

design constraints

design specifications

schematic generation

simulation and characterisation

specification met? release to

manufacturing

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General linear methods for integrated circuit design – St. Voigtmann, p. 2

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Classical methods

I BDF

. artificial damping

-1 0 1

20 time 10 0

I Trapezoidal rule

. undesired oscillations

2

1 0 1 2

time π 2 π 0 2

I Runge-Kutta methods . high computational costs

0 30 60 90

problem size 30

20 10 0

computingtime

Runge Kutta BDF

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Classical methods

I BDF

. artificial damping

-1 0 1

20 time 10 0

I Trapezoidal rule

. undesired oscillations

2

1 0 1 2

time π 2 π 0 2

I Runge-Kutta methods . high computational costs

0 30 60 90

problem size 30

20 10 0

computingtime

Runge Kutta BDF

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General linear methods for integrated circuit design – St. Voigtmann, p. 2

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Classical methods

I BDF

. artificial damping

-1 0 1

20 time 10 0

I Trapezoidal rule

. undesired oscillations

2

1 0 1 2

time π 2 π 0 2

I Runge-Kutta methods . high computational costs

0 30 60 90

problem size 30

20 10 0

computingtime

Runge Kutta BDF

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Classification of methods

I Linear multistep methods . low costs

. very successful (BDF) . not A-stable forp>2 I Runge-Kutta methods

. very good stability properties . stepsize change is easy . high costs

I General linear methods (GLM) . combine advantages

of both classes

. make new methods possible . provide unifying framework

for known methods

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General linear methods for integrated circuit design – St. Voigtmann, p. 4

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Contents

Differential Algebraic Equations

General Linear Methods

Practical General Linear Methods

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs of increasing complexity

A[Dx]0+Bx=q A[Dx]0+b(x,·) =0 A d0(x,·)+b(x,·) =0

Index2 Index1 Index0

ODEs

lin. DAEs

Hessenberg

nonlin. index-1DAEs circuit simulation

prop. stated index-2DAEs

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General linear methods for integrated circuit design – St. Voigtmann, p. 5

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs of increasing complexity

A[Dx]0+Bx=q A[Dx]0+b(x,·) =0 A d0(x,·)+b(x,·) =0

Mx0=f(x,·)

Index2 Index1

Index0 ODEs

lin. DAEs

Hessenberg

nonlin. index-1DAEs circuit simulation

prop. stated index-2DAEs

y0 =f(y,t) ordinary differential equations

I well understood (theoretically, numerically)

I Butcher, Dahlquist, Gear, Hairer, Petzold, . . .

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs of increasing complexity

A[Dx]0+Bx=q A[Dx]0+b(x,·) =0 A d0(x,·)+b(x,·) =0

Mx0=f(x,·)

Index2 Index1

Index0 ODEs

lin. DAEs

Hessenberg

nonlin. index-1DAEs circuit simulation

prop. stated index-2DAEs

E x0+Fx=q A[Dx]0+Bx=q

linear DAEs

I standard form (Hairer/Wanner, Kunkel/Mehrmann)

I prop. stated (M ¨arz, Balla, Kurina)

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General linear methods for integrated circuit design – St. Voigtmann, p. 5

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs of increasing complexity

A[Dx]0+Bx=q A[Dx]0+b(x,·) =0 A d0(x,·)+b(x,·) =0

Mx0=f(x,·)

Index2 Index1

Index0 ODEs

lin. DAEs

Hessenberg

nonlin. index-1DAEs circuit simulation

prop. stated index-2DAEs

y0=f(y,z) 0=g(z)

DAEs in Hessenberg form

I Runge-Kutta (Hairer/Wanner, Kværnø)

I lin. multistep (Campbell, Gear, Petzold)

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs of increasing complexity

A[Dx]0+Bx=q A[Dx]0+b(x,·) =0 A d0(x,·)+b(x,·) =0

Mx0=f(x,·)

Index2 Index1

Index0 ODEs

lin. DAEs

Hessenberg

nonlin. index-1DAEs

circuit simulation

prop. stated index-2DAEs

A d0(x,·)+b(x,·) =0 nonlinear index-1 DAEs (prop. stated)

I extension of decoupling procedure

I M ¨arz, Higueras

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General linear methods for integrated circuit design – St. Voigtmann, p. 5

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs of increasing complexity

A[Dx]0+Bx=q A[Dx]0+b(x,·) =0 A d0(x,·)+b(x,·) =0

Mx0=f(x,·)

Index2 Index1

Index0 ODEs

lin. DAEs

Hessenberg

nonlin. index-1DAEs circuit simulation

prop. stated index-2DAEs

A d0(x,·)+b(Ux,·) +BTx=0

DAEs appearing in electrical circuit simulation

I index-2, no Hessenberg form

I Tischendorf, Est ´evez Schwarz (initialisation)

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs of increasing complexity

A[Dx]0+Bx=q A[Dx]0+b(x,·) =0 A d0(x,·)+b(x,·) =0

Mx0=f(x,·)

Index2 Index1

Index0 ODEs

lin. DAEs

Hessenberg

nonlin. index-1DAEs circuit simulation

prop. stated index-2DAEs A d0(x,·)+b(x,·) =0 nonlinear DAEs with properly stated leading terms

I existence and uniqueness of solutions

I convergence results for numerical methods

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General linear methods for integrated circuit design – St. Voigtmann, p. 6

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs in electrical circuit simulation

I Modified Nodal Analysis Aq(x,˙ ·)+b(x,·) =0

I analysis: tractability index . low smoothness requirements . use projectors (Pi,Qi,U,T, . . . )

and matrix sequences

I index can be determined topologically . look for CV loops and LI cutsets I index-2componentsTxare given by

. currents ofV-sources inCVloops . voltages of inductors andI-sources

inLIcutsets

I index-2componentsTxenter linearly Aq(x,˙ ·)+b(Ux,·) +BTx=0

V1

V2

VBB

VDD u4

u12 u1

u3

u2

u7

u6 u5

u11

u10

u9 u8

C

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs in electrical circuit simulation

I Modified Nodal Analysis Aq(x,˙ ·)+b(x,·) =0

I analysis: tractability index . low smoothness requirements . use projectors (Pi,Qi,U,T, . . . )

and matrix sequences

I index can be determined topologically . look for CV loops and LI cutsets I index-2componentsTxare given by

. currents ofV-sources inCVloops . voltages of inductors andI-sources

inLIcutsets

I index-2componentsTxenter linearly Aq(x,˙ ·)+b(Ux,·) +BTx=0

V1

V2

VBB

VDD u4

u12 u1

u3

u2

u7

u6 u5

u11

u10

u9 u8

C

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General linear methods for integrated circuit design – St. Voigtmann, p. 6

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs in electrical circuit simulation

Est ´evez Schwarz (2000) Cons. initialization for index-2 DAEs and it’s application to circuit simulation, PhD thesis

I Modified Nodal Analysis Aq(x,˙ ·)+b(x,·) =0

I analysis: tractability index . low smoothness requirements . use projectors (Pi,Qi,U,T, . . . )

and matrix sequences

I index can be determined topologically . look for CV loops and LI cutsets I index-2componentsTxare given by

. currents ofV-sources inCVloops . voltages of inductors andI-sources

inLIcutsets

I index-2componentsTxenter linearly Aq(x,˙ ·)+b(Ux,·) +BTx=0

V1

V2

VBB

VDD u4

u12 u1

u3

u2

u7

u6 u5

u11

u10

u9 u8

C

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs in electrical circuit simulation (cont.)

A[Dx]0+b(Ux,·) +BTx=0 ➪ index-2 components enter linearly

I Idea: Introduce new variables

u=DP1x, w=P1D(Dx)0+ (Q0+Q1)x.

For a solutionxthis implies

x=Du+ (P0Q1+Q0P1)w+Q0Q1D(Dx)0.

I Consequences:

F(u,w,·) =A[Dx]0+b(Ux,·) +BTx=0.

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General linear methods for integrated circuit design – St. Voigtmann, p. 7

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs in electrical circuit simulation (cont.)

A[Dx]0+b(Ux,·) +BTx=0 ➪ index-2 components enter linearly

I Idea: Introduce new variables

u=DP1x, w=P1D(Dx)0+ (Q0+Q1)x.

For a solutionxthis implies

x=Du+ (P0Q1+Q0P1)w+Q0Q1D(Dx)0.

I Consequences:

F(u,w,·) =A[Dx]0+b(Ux,·) +BTx=0.

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## DAEs in electrical circuit simulation (cont.)

A[Dx]0+b(Ux,·) +BTx=0 ➪ index-2 components enter linearly

I Idea: Introduce new variables

u=DP1x, w=P1D(Dx)0+ (Q0+Q1)x.

For a solutionxthis implies

x=Du+ (P0Q1+Q0P1)w+Q0Q1D(Dx)0.

I Consequences:

F(u,w,·) =A[Dx]0+b(Ux,·) +BTx=0.

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General linear methods for integrated circuit design – St. Voigtmann, p. 8

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Local existence and uniqueness of solutions

V. (2004) General linear methods for nonlinear DAEs in circuit simulation, SCEE

Theorem.

I The properly stated index-2DAE

F(u,w,·) =A[Dx]0+b(Ux,·) +BTx=0, F(u0,w0,t0) =0,

is locally equivalent to w(u0,t0) =w0, F u,w(u,t),t

=0.

I For everyx0∈IRm, the initial value problem

A[Dx]0+b(Ux,·) +BTx=0, DP1x(t0) =DP1x0.

is uniquely solvable. The solutionx=Du+z0+z1satisfies u0 =f u,w(u,t),t

, z1=P0Q1w(u,t), z0 =g u,(Dz1)0,t .

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Local existence and uniqueness of solutions

Theorem.

I The properly stated index-2DAE

F(u,w,·) =A[Dx]0+b(Ux,·) +BTx=0, F(u0,w0,t0) =0,

is locally equivalent to w(u0,t0) =w0, F u,w(u,t),t

=0.

I For everyx0∈IRm, the initial value problem

A[Dx]0+b(Ux,·) +BTx=0, DP1x(t0) =DP1x0.

is uniquely solvable. The solutionx=Du+z0+z1satisfies u0 =f u,w(u,t),t

, z1=P0Q1w(u,t), z0 =g u,(Dz1)0,t .

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General linear methods for integrated circuit design – St. Voigtmann, p. 8

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Local existence and uniqueness of solutions

V. (2004) General linear methods for nonlinear DAEs in circuit simulation, SCEE

Theorem.

I The properly stated index-2DAE

F(u,w,·) =A[Dx]0+b(Ux,·) +BTx=0, F(u0,w0,t0) =0,

is locally equivalent to w(u0,t0) =w0, F u,w(u,t),t

=0.

I For everyx0∈IRm, the initial value problem

A[Dx]0+b(Ux,·) +BTx=0, DP1x(t0) =DP1x0.

is uniquely solvable. The solutionx=Du+z0+z1satisfies u0 =f u,w(u,t),t

, z1=P0Q1w(u,t), z0 =g u,(Dz1)0,t

➩ .

inherent ordinary differential equation

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Properly stated index-2DAEs

### Aq(x, ˙ ·)+ b(x, ·) = 0

I split solution into characteristic parts x=Du+z+w

x

P0x z0=Q0x

DDP¯1x z1=P0Q¯1x U Q0x T Q0x Du z=z1+U z0 w=T z0

I split equations similarly u0=f(u,v0,t)

v=g(u,t)

. x =Du+z(u,·) +w(u,v0,·) z=z(u,t) w=w(u,v0,t)

F(u, w, z, η, ζ, t) = 0

Fˆ1(u, z, t) = 0 u0= (u, w, t) Fˆ2(u, w, ζ, t) = 0

z=(u, t)

v=(u, t) w=(u, ζ, t)

u0=f(u, v0, t) v=g(u, t)

¯ ZG¯−1

2 P1G¯−1

2 G−1

2

I Ifv0gu remains non-singular (locally) Implicit Index-1 System

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General linear methods for integrated circuit design – St. Voigtmann, p. 9

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Properly stated index-2DAEs

V. (2004) Accessible criteria for the local existence and uniqueness of DAE solutions, MATHEON

### Aq(x, ˙ ·)+ b(x, ·) = 0

I split solution into characteristic parts x=Du+z+w

x

P0x z0=Q0x

DDP¯1x z1=P0Q¯1x U Q0x T Q0x

Du z=z1+U z0 w=T z0

I split equations similarly u0=f(u,v0,t)

v=g(u,t)

. x =Du+z(u,·) +w(u,v0,·) z=z(u,t) w=w(u,v0,t)

F(u, w, z, η, ζ, t) = 0

Fˆ1(u, z, t) = 0 u0= (u, w, t) Fˆ2(u, w, ζ, t) = 0

z=(u, t)

v=(u, t) w=(u, ζ, t)

u0=f(u, v0, t) v=g(u, t)

¯ ZG¯−1

2 P1G¯−1

2 G−1

2

I Ifv0gu remains non-singular (locally) Implicit Index-1 System

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Properly stated index-2DAEs

### Aq(x, ˙ ·)+ b(x, ·) = 0

I split solution into characteristic parts x=Du+z+w

x

P0x z0=Q0x

DDP¯1x z1=P0Q¯1x U Q0x T Q0x

Du z=z1+U z0 w=T z0

I split equations similarly u0=f(u,v0,t)

v=g(u,t)

. x =Du+z(u,·) +w(u,v0,·) z=z(u,t) w=w(u,v0,t)

F(u, w, z, η, ζ, t) = 0

Fˆ1(u, z, t) = 0 u0= (u, w, t) Fˆ2(u, w, ζ, t) = 0

z=(u, t)

v=(u, t) w=(u, ζ, t)

u0=f(u, v0, t) v=g(u, t)

¯ ZG¯−1

2 P1G¯−1

2 G−1

2

I Ifv0gu remains non-singular (locally) Implicit Index-1 System

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General linear methods for integrated circuit design – St. Voigtmann, p. 10

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Properly stated index-2DAEs (cont.)

V. (2006) General linear methods for integrated circuit design, PhD thesis

### Aq(x, ˙ ·)+ b(x, ·) = 0

u0=f(u,v0,t) v=g(u,t)

x=Du+z(u,·) +w(u,v0,·) z=z(u,t) w=w(u,v0,t)

I new decoupling procedure

I existence and uniqueness results

I only mild smoothness assumptions

I covers/extends results on

. linear DAEs (Balla, M ¨arz, Kurina) . nonlinear index-1 DAEs (Higueras, M ¨arz)

. DAEsA[Dx]0+b(Ux,·) +BTx=0 (Tischendorf, Est ´evez Schwarz) . Hessenberg DAEs (Hairer, Lubich, Roche, Wanner)

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Properly stated index-2DAEs (cont.)

### Aq(x, ˙ ·)+ b(x, ·) = 0

u0=f(u,v0,t) v=g(u,t)

x=Du+z(u,·) +w(u,v0,·) z=z(u,t) w=w(u,v0,t)

I new decoupling procedure

I existence and uniqueness results

I only mild smoothness assumptions

I covers/extends results on

. linear DAEs (Balla, M ¨arz, Kurina) . nonlinear index-1 DAEs (Higueras, M ¨arz)

. DAEsA[Dx]0+b(Ux,·) +BTx=0 (Tischendorf, Est ´evez Schwarz) . Hessenberg DAEs (Hairer, Lubich, Roche, Wanner)

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General linear methods for integrated circuit design – St. Voigtmann, p. 11

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Decoupling and discretisation

Aq(x,˙ t) +b(x,t) =0 index-2 DAE

decoupling u0=f(u,v0,t) v=g(u,t) +constraints

discretisation discretisation

discretised index-2 DAE

decoupling discretised index-1 DAE +discretised constraints

I If two subspaces associated with the DAE,DN1andDS1are constant, then this diagram commutes.

I It is always assumed thatN0S0does not depend onx.

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Decoupling and discretisation

Aq(x,˙ t) +b(x,t) =0 index-2 DAE

decoupling u0=f(u,v0,t) v=g(u,t) +constraints

discretisation discretisation

discretised index-2 DAE

decoupling discretised index-1 DAE +discretised constraints

I If two subspaces associated with the DAE,DN1andDS1are constant, then this diagram commutes.

I It is always assumed thatN0S0does not depend onx.

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General linear methods for integrated circuit design – St. Voigtmann, p. 11

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Decoupling and discretisation

Aq(x,˙ t) +b(x,t) =0 index-2 DAE

decoupling u0=f(u,v0,t) v=g(u,t) +constraints

discretisation discretisation

discretised index-2 DAE

decoupling discretised index-1 DAE +discretised constraints

I If two subspaces associated with the DAE,DN1andDS1are constant, then this diagram commutes.

I It is always assumed thatN0S0does not depend onx.

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Contents

Differential Algebraic Equations

General Linear Methods

Practical General Linear Methods

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General linear methods for integrated circuit design – St. Voigtmann, p. 13

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

0

## = f (y)

I Linear multistep: yn = hβ0f(yn) + Pk

i=1αiyn−i

I Runge-Kutta: Yi = hPs

j=1aijf(Yj) + y[n−1], y[n] = hPs

i=1 bif(Yi) + y[n−1]

I General linear: Yi = hPs

j=1aijf(Yj) + Pr

j=1uijy[n−1]j , y[n]i = hPs

j=1 bijf(Yj) + Pr

j=1vijy[n−1]j

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

0

## = f (y)

I Linear multistep: yn = hβ0f(yn) + Pk

i=1αiyn−i

I Runge-Kutta: Yi = hPs

j=1aijf(Yj) + y[n−1], y[n] = hPs

i=1 bif(Yi) + y[n−1]

I General linear: Yi = hPs

j=1aijf(Yj) + Pr

j=1uijy[n−1]j , y[n]i = hPs

j=1 bijf(Yj) + Pr

j=1vijy[n−1]j

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General linear methods for integrated circuit design – St. Voigtmann, p. 13

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

0

## = f (y)

I Linear multistep: yn = hβ0f(yn) + Pk

i=1αiyn−i I Runge-Kutta: Yi = hPs

j=1aijf(Yj) + y[n−1], y[n] = hPs

i=1 bif(Yi) + y[n−1]

I General linear: Yi = hPs

j=1aijf(Yj) + Pr

j=1uijy[n−1]j , y[n]i = hPs

j=1 bijf(Yj) + Pr

j=1vijy[n−1]j

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

0

## = f (y)

I Linear multistep: yn = hβ0f(yn) + Pk

i=1αiyn−i I Runge-Kutta: Yi = hPs

j=1aijf(Yj) + y[n−1], y[n] = hPs

i=1 bif(Yi) + y[n−1]

I General linear: Yi = hPs

j=1aijf(Yj) + Pr

j=1uijy[n−1]j , y[n]i = hPs

j=1 bijf(Yj) + Pr

j=1vijy[n−1]j

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General linear methods for integrated circuit design – St. Voigtmann, p. 13

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

0

## = f (y)

I Linear multistep: yn = hβ0f(yn) + Pk

i=1αiyn−i I Runge-Kutta: Yi = hPs

j=1aijf(Yj) + y[n−1], y[n] = hPs

i=1 bif(Yi) + y[n−1]

I General linear: Yi = hPs

j=1aijf(Yj) + Pr

j=1uijy[n−1]j , y[n]i = hPs

j=1 bijf(Yj) + Pr

j=1vijy[n−1]j

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## An example method

"

A U B V

#

=

1

4 0 0 0 1 0 −3211921

49 25

1

4 0 0 1−1711001004960043

123

1225225392 14 0 1 13631400 1394139200 784005379

95845984 367 14 1 4318 3142 33637

95845984 367 14 1 4318 3142 33637

0 0 0 1 0 0 0 0

26821 8621289 4 0 709 1021215

3221 88212249 16 0 569 20211021

(Butcher, 2004) I diagonally implicit

I orderp=3and stage orderq=3

I stiffly accurate, A-stable, L-stable

I Nordsieck form, y[n]i+1hiy(i)(tn)

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General linear methods for integrated circuit design – St. Voigtmann, p. 14

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## An example method

"

A U B V

#

=

1

4 0 0 0 1 0 −3211921

49 25

1

4 0 0 1−1711001004960043

123

1225225392 14 0 1 13631400 1394139200 784005379

95845984 367 14 1 4318 3142 33637

95845984 367 14 1 4318 3142 33637

0 0 0 1 0 0 0 0

26821 8621289 4 0 709 1021215

3221 88212249 16 0 569 20211021

(Butcher, 2004) I diagonally implicit

I orderp=3and stage orderq=3

I stiffly accurate, A-stable, L-stable

I Nordsieck form, y[n]i+1hiy(i)(tn)

Diagonally implicit methods with high stage order

are possible!

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Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Why general linear methods?

"

Y y[n]

#

=

"

A U B V

#

·

"

hf(Y) y[n−1]

#

I improve stability . damping properties

similar to RK methods -1

0 1

20 time 10

0 2

1 0 1 2

time π 2 π 0 2

I improve efficiency

. diagonally implicit schemes . solve stages sequentially

A=

λ 0

... . .. aij · · · λ

I benefit from high stage order . no order reduction

. cheap and reliable error estimates

1.0·10−2 1.5·10−2

2.0·10−2

0 5 10 15 time

p= 3 p= 2 p= 1

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General linear methods for integrated circuit design – St. Voigtmann, p. 15

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Why general linear methods?

"

Y y[n]

#

=

"

A U B V

#

·

"

hf(Y) y[n−1]

#

I improve stability . damping properties

similar to RK methods -1

0 1

20 time 10

0 2

1 0 1 2

time π 2 π 0 2

I improve efficiency

. diagonally implicit schemes . solve stages sequentially

A=

λ 0

... . ..

aij · · · λ

I benefit from high stage order . no order reduction

. cheap and reliable error estimates

1.0·10−2 1.5·10−2

2.0·10−2

0 5 10 15 time

p= 3 p= 2 p= 1

(42)

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## Why general linear methods?

"

Y y[n]

#

=

"

A U B V

#

·

"

hf(Y) y[n−1]

#

I improve stability . damping properties

similar to RK methods -1

0 1

20 time 10

0 2

1 0 1 2

time π 2 π 0 2

I improve efficiency

. diagonally implicit schemes . solve stages sequentially

A=

λ 0

... . ..

aij · · · λ

I benefit from high stage order . no order reduction

. cheap and reliable error estimates

1.0·10−2 1.5·10−2

2.0·10−2

0 5 10 15 time

p= 3 p= 2 p= 1

(43)

General linear methods for integrated circuit design – St. Voigtmann, p. 16

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## GLMs for index-2 DAEs

A q x(t),t 0+b x(t),t

=0

| {z }

singular charges/ voltages/

fluxes currents

I input quantities

q[n−1]k+1hk dd tkkq x(t),t

I

"

q(Xn,tc) q[n]

#

= A U

B V

·

" hQ0n q[n−1]

#

such that

AQ0n+b(Xn,tc) =0

I solve for the stagesXn Remark

I use implicit methods (Anon-singular)

I charge conservation is guaranteed

I only charges / fluxes are passed on from step to step

I analysis usesimplicit index-1 systemy0=f(y,z0), z=g(y)

(44)

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## GLMs for index-2 DAEs

A q x(t),t 0+b x(t),t

=0

| {z }

singular charges/ voltages/

fluxes currents

I input quantities

q[n−1]k+1hk dd tkkq x(t),t

I

"

q(Xn,tc) q[n]

#

= A U

B V

·

"

hQ0n q[n−1]

#

such that

AQ0n+b(Xn,tc) =0

I solve for the stagesXn

Remark

I use implicit methods (Anon-singular)

I charge conservation is guaranteed

I only charges / fluxes are passed on from step to step

I analysis usesimplicit index-1 systemy0=f(y,z0), z=g(y)

(45)

General linear methods for integrated circuit design – St. Voigtmann, p. 16

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## GLMs for index-2 DAEs

A q x(t),t 0+b x(t),t

=0

| {z }

singular charges/ voltages/

fluxes currents

I input quantities

q[n−1]k+1hk dd tkkq x(t),t

I

"

q(Xn,tc) q[n]

#

= A U

B V

·

"

hQ0n q[n−1]

#

such that

AQ0n+b(Xn,tc) =0

I solve for the stagesXn Remark

I use implicit methods (Anon-singular)

I charge conservation is guaranteed

I only charges / fluxes are passed on from step to step

I analysis usesimplicit index-1 systemy0=f(y,z0), z=g(y)

(46)

Motivation DAEs General Linear Methods Practical General Linear Methods Summary

## GLMs for index-1 DAEs

Apply M=

A U

B V

to implicit index-1 DAEs y0=f(y,z0), z=g(y) Y=hAf(Y,Z0) + Uy[n] g(Y) =hAZ0+Uz[n]

y[n+1]=hBf(Y,Z0) +Vy[n] z[n+1]=hBZ0+ Vz[n]

Updating...

## References

Related subjects :