Large Timestep Issues

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Large Timestep Issues

• Lecture 12
• Alessandra Nardi

Thanks to Prof. Sangiovanni, Prof. Newton, Prof.
White, Deepak Ramaswamy, Michal Rewienski, and
Karen Veroy
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Last lecture review

• Transient Analysis of dynamical circuits
• i.e., circuits containing C and/or L
• Examples
• Solution of ODEs (IVP)
• Forward Euler (FE), Backward Euler (BE) and
  Trapezoidal Rule (TR)
• Multistep methods
• Convergence
• Consistency

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Outline

• Convergence for multistep methods
• Stability
• Region of Absolute Stability
• Dahlquists Stability Barriers
• Stiff Stability (Large timestep issues)
• Examples
• Analysis of FE, BE
• Gears Method
• Variable step size
• More on Implicit Methods
• Solution with NR
• Application of multistep to circuit equations

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Multistep Methods Common AlgorithmsTR, BE, FE
are one-step methods
Multistep Equation
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Multistep 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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Multistep Methods StabilityDifference Equation
Why did the best 2-step explicit method fail to
Converge?
Multistep Method Difference Equation
LTE
Global Error
We made the LTE so small, how come the Global
error is so large?
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An Aside on Solving Difference Equations
Consider a general kth order difference equation
Three important observations
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Multistep Methods StabilityDifference Equation
Multistep Method Difference Equation
Definition A multistep method is stable if and
only if
Theorem A multistep method is stable if and only
if
Less than one in magnitude or equal to one and
distinct
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Multistep Methods StabilityStability Theorem
Proof
Given the Multistep Method Difference Equation
are either
If the roots of

• less than one in magnitude
• equal to one in magnitude but distinct

Then from the aside on difference equations
From which stability easily follows.
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Multistep Methods StabilityStability Theorem
Proof
Im
Re
1
-1
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Multistep Methods StabilityA more formal
approach

• Def A method is stable if all the solutions of
  the associated difference equation obtained from
  (1) setting q0 remain bounded if l??
• The region of absolute stability of a method is
  the set of q such that all the solutions of (1)
  remain bounded if l??
• Note that a method is stable if its region of
  absolute stability contains the origin (q0)

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Multistep Methods StabilityA more formal
approach
Def A method is A-stable if the region of
absolute stability contains the entire left hand
plane (in the ? space)

Im(?)
Re(?)
-1
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Multistep Methods StabilityA more formal
approach

• Each method is associated with two polynomials a
  and b
• a associated with function past values
• b associated with derivative past values
• Stability roots of a must stay in z?1 and be
  simple on z1
• Absolute stability roots of (a-bq) must stay in
  z?1 and be simple on z1 when Re(q)lt0.

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Multistep Methods StabilityDahlquists
Stability Barriers

• First For a stable, explicit k-step multistep
  method, the maximum number of exactness
  constraints that can be satisfied is less than or
  equal to k (note there are 2k coefficients). For
  implicit methods, the number of constraints that
  can be satisfied is either k2 if k is even or
  k1 if k is odd.
• Second There are no A-stable methods of
  convergence order greater than 2, and the
  trapezoidal rule is the most accurate.

TR very popular (SPICE)
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Multistep Methods Convergence
AnalysisConditions for convergence Consistency
Stability
1) Local Condition One step errors are small
(consistency)
Exactness Constraints up to p0 (p0 must be gt 0)
2) Global Condition One step errors grow slowly
(stability)
Convergence Result
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Multistep MethodsFE region of absolute stability
Forward Euler
ODE stability region
Im(z)
Difference Eqn Stability region
Region of Absolute Stability
Re(z)
1
-1
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Multistep MethodsBE region of absolute stability
Backward Euler
Im(z)
Difference Eqn Stability region
Re(z)
1
-1
Region of Absolute Stability
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Summary

• Convergence for one-step methods
• Consistency for FE
• Stability for FE
• Convergence for multistep methods
• Consistency (Exactness Constraints)
• Selecting coefficients
• Stability
• Region of Absolute Stability
• Dahlquists Stability Barriers

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Stiff Problems (Large Timestep Issues)Example

• Interval of interest is 0,5
• Uniform step size (for accuracy)
• ? Dt ? 10-6
• ? 5x106 steps !!!

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Stiff Problems (Large Timestep Issues)Example

• Strategy (for previous example) Take 5 steps of
  size
• 10-6 for accuracy during initial phase and then 5
  steps of
• size 1.
• Stiff problem
• Natural time constants
• Input time constants
• Interval of interest
• If these are widely separated, then the problem
  is stiff

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Application ProblemsSignal Transmission in an IC
2x2 example
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Stiff Problems (Large Timestep Issues)FE on two
time-constant 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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Stiff Problems (Large Timestep Issues)BE on two
time-constant circuit
Circuit Example
Backward-Euler Computed Solution
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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Multistep Methods (Large Timestep Issues)BE, FE,
TR on the scalar ODE problem
Scalar ODE
Forward-Euler
Backward-Euler
Trap Rule
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Stiff Problems (Large Timestep Issues)FE on two
time-constant circuit
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Stiff Problems (Large Timestep Issues)BE on two
time-constant circuit
Region of Absolute Stability
Region of Absolute Stability
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Stiff Problems

• We showed that
• The analysis of stiff circuits requires the use
  of variable step sizes
• Not all the linear multistep methods can be
  efficiently used to integrate stiff equations
• To be able to choose Dt based only on accuracy
  considerations, the region of absolute stability
  should allow a large Dt for large time constants,
  without being constrained by the small time
  constants
• Clearly A-stable methods satisfy this requirement

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Backward Differentiation Formula - BDF (Gear
Methods)

• Note that Gears first order method is BE
• It can be shown that
• Gears methods up to order 6 are stiffly stable
  and are well-suited for stiff ODEs
• Gears methods of order higher than 6 are not
  stiffly stable
• Less stringent than A-stable

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Gears Method region of absolute
stability(outside the closed curve)
k1
k2
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Gears Method region of absolute
stability(outside the closed curve)
k3
k4
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Variable step size

• When the step size is changed during the
  integration, the coefficients of the method need
  to be recomputed at each iteration
• Example Gears method of order 2

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Variable step size
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More observations

• To minimize the computation time needed to
  integrate differential equations, the Dt must be
  chosen as large as possible provided that the
  desired accuracy is achieved
• Several approximation are available. SPICE2 uses
  Divided Differences
• At a certain time point, different integration
  methods would allow different step size
• Advantageous to implement a strategy which allows
  a change of method as well as of Dt

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Summary on Stiff Stability

• FE timestep is limited by stability and not by
  accuracy
• BE A-stable, any timestep could be used
• TR most accurate A-stable multistep method
• Gear stiffly stable method (up to order 6)
• The analysis of stiff circuits requires the use
  of variable timestep

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Multistep MethodsMore on Implicit Methods
Forward-Euler
Backward-Euler
Requires just function Evaluations
Nonlinear equation solution at each step
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Multistep Implicit MethodsSolution with Newton
Rewrite the multistep Equation
Solve with Newton
Here j is the Newton iteration index
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Multistep Implicit MethodsSolution with Newton
Newton Iteration
Solution with Newton is very efficient
Easy to generate a good initial guess using
polynomial fitting
Jacobian become easy to factor for small timesteps
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Application of linear multistep methods to
circuit equations
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Transient Analysis Flow Diagram
Predict values of variables at tl
Replace C and L with resistive elements via
integration formula
Replace nonlinear elements with G and indep.
sources via NR
Assemble linear circuit equations
Solve linear circuit equations
NO
Did NR converge?
YES
Test solution accuracy
Save solution if acceptable
Select new Dt and compute new integration formula
coeff.
NO
Done?
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Summary

• Transient Analysis of dynamical circuits
• Solution of ODEs (IVP)
• FE, BE and TR
• Multistep methods
• Convergence
• Consistency
• Stability
• Stiff Stability (Large timestep issues)
• Gears method
• Application of multistep to circuit equations
• Did not talk about
• Runge-Kutta
• Predictor-Corrector Methods

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Summary on circuit simulation

• Circuit Equation Formulation
• STA, MNA
• DC Analysis of Nonlinear Circuits
• Solution of Linear Equations (direct and
  iterative methods)
• Solution of Nonlinear Equations (Newtons
  method)
• Transient Analysis of Nonlinear Circuits
• Solution of Ordinary Differential Equations- IVP
  (multistep methods)

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Appendix to circuit simulation Preconditioners

• The convergence rate of iterative methods depends
  on spectral properties of the coefficient matrix.
  Hence one may attempt to transform the linear
  system into one that is equivalent, but that has
  more favorable spectral properties.
• A preconditioner is a matrix that effects such a
  transformation
• Mxb ? A-1MxA-1b
• The choice of a preconditioner is largely
  application specific