1. Lecture 11 Alessandra Nardi
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 Examples
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
Dahlquist’s Stability Barriers

4. Multistep Methods – Common Algorithms TR, BE, FE are one-step methods
Multistep Equation:
FE Discrete Equation:
Forward-Euler Approximation:
Multistep Coefficients:
Multistep Coefficients:
BE Discrete Equation:
Trap Discrete Equation:
Multistep Coefficients:

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.
multistep
discretization
Decoupled Equations

20. Multistep Methods Simplified Problem for Analysis
Scalar ODE:
Scalar Multistep formula:
Decaying
Solutions
Osci l
lations
Growing
Solutions

21. Multistep Methods – Convergence Analysis Global Error Equation
Multistep formula:
Exact solution Almost
satisfies Multistep Formula:
Local Truncation Error (LTE)
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 ConstraintsIf
As any smooth v(t) has a locally accurate Taylor series in t: if
Then

24. For k=2, yields a 5x6 system of equations for Coefficients
Multistep Methods – Making LTE small Exactness Constraints – k=2 Example
For k=2, yields a 5x6 system of equations for Coefficients
p=0
p=1
p=2
p=3
p=4

25. Exactness Constraints for k=2
Multistep Methods – Making LTE small Exactness Constraints – k=2 Example
Exactness Constraints for k=2

26. Multistep Methods – Making LTE small Exactness Constraints k=2 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 small10 -4 -3 -2 -1
-15
-10 -5 FE
LTE
Trap
Best Explicit Method has highest one-step
accurate
Beste
Timestep

28. Multistep Methods – Making LTE small
Max Error FE
Where’s BESTE?
Trap
Timestep

29. Multistep Methods – Making LTE small
worrysome 10 -4 -3 -2 -1
-100
100
200
Max Error
Best Explicit Method has lowest one-step error but global errror increases as timestep decreases
Beste
Trap FE
Timestep

30. Multistep Methods – Stability Difference 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 – Stability Difference 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 – Stability Stability 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 – Stability Stability Theorem ProofIm Re -1 1

37. Multistep Methods – Stability A more formal approach
Def: A method is stable if all the solutions of the associated difference equation obtained from (1) setting q=0 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 (q=0)

38. Multistep Methods – Stability A more formal approach
Def: A method is A-stable if the region of absolute stability contains the entire left hand plane (in the  space)
Re(z)
Im(z) -1 1
Im()
Re() -1 1

39. Multistep Methods – Stability A 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 |z|=1
Absolute stability: roots of (a-bq) must stay in |z|1 and be simple on |z|=1 when Re(q)<0.

40. Multistep Methods – Stability Dahlquist’s 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 k+2 if k is even or k+1 if k is odd.

41. Multistep Methods – Convergence Analysis Conditions for convergence – Consistency & Stability
1) Local Condition: One step errors are small (consistency)
Exactness Constraints up to p0 (p0 must be > 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
Dahlquist’s Stability Barriers