Multistep Methods

1
Multistep Methods

• Lecture 11
• Alessandra Nardi

Thanks to Prof. Jacob White, Deepak Ramaswamy,
Michal Rewienski, and Karen Veroy
2
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
• Stability

3
Outline

• 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

4
Multistep Methods Common AlgorithmsTR, BE, FE
are one-step methods
Multistep Equation
5
Multistep Methods Convergence Analysis
Convergence Definition
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
6
Multistep Methods Convergence Analysis Order-p
Convergence
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
Forward- and Backward-Euler are order 1 convergent
Trapezoidal Rule is order 2 convergent
7
Multistep Methods Convergence Analysis Two
types of error
8
Multistep Methods Convergence Analysis Two
conditions for Convergence

• 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 Dt, 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.

9
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.
10
One-step Methods Convergence Analysis
Consistency definition
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
11
One-step Methods Convergence Analysis
Consistency for Forward Euler
Proves the theorem if derivatives of x are bounded
12
One-step Methods Convergence Analysis
Convergence Analysis for Forward Euler
13
One-step Methods Convergence Analysis
Convergence Analysis for Forward Euler
14
One-step Methods Convergence Analysis A
helpful bound on difference equations
15
One-step Methods Convergence Analysis A
helpful bound on difference equations
16
One-step Methods Convergence Analysis Back to
Convergence Analysis for Forward Euler
17
One-step Methods Convergence Analysis
Observations about Convergence Analysis for FE

• 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).

18
Multistep Methods Definition and Observations
Multistep Equation
How does one pick good coefficients?
Want the highest accuracy
19
Multistep Methods Simplified Problem for Analysis
Scalar ODE
Why such a simple Test Problem?

• Nonlinear Analysis has many unrevealing
  subtleties
• Scalar equivalent to vector for multistep methods.

20
Multistep Methods Simplified Problem for Analysis
Scalar ODE
Scalar Multistep formula
Growing Solutions
Decaying Solutions
21
Multistep Methods Convergence Analysis Global
Error Equation
Multistep formula
Exact solution Almost satisfies Multistep Formula
Global Error
Subtracting yields difference equation for global
error
22
Multistep Methods Making LTE small Exactness
Constraints
Multistep methods are designed so that they are
exact for a polynomial of order p. These methods
are said to be of order p.
23
Multistep Methods Making LTE small Exactness
Constraints
As any smooth v(t) has a locally accurate Taylor
series in t
Then
24
Multistep Methods Making LTE small Exactness
Constraints k2 Example
For k2, yields a 5x6 system of equations for
Coefficients
p0
p1
p2
p3
p4
25
Multistep Methods Making LTE small Exactness
Constraints k2 Example
Exactness Constraints for k2
26
Multistep Methods Making LTE small Exactness
Constraintsk2 Example, generating Methods
Solve for the 2-step method with lowest LTE
Solve for the 2-step explicit method with lowest
LTE
27
Multistep Methods Making LTE small
FE
LTE
Trap
Beste
Timestep
28
Multistep Methods Making LTE small
Max Error
FE
Wheres BESTE?
Trap
Timestep
29
Multistep Methods Making LTE small
Max Error
Beste
Trap
FE
Timestep
30
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?
31
An Aside on Solving Difference Equations
Consider a general kth order difference equation
Which must have k initial conditions
As is clear when the equation is in update form
Most important difference equation result
32
An Aside on Solving Difference Equations
To understand how h is derived, first a simple
case
33
An Aside on Solving Difference Equations
Three important observations
34
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
35
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.
36
Multistep Methods StabilityStability Theorem
Proof
Im
Re
1
-1
37
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)

38
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
1
39
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.

40
Multistep Methods StabilityDahlquists First
Stability Barrier
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.
41
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
42
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