Lecture 11 Alessandra Nardi Multistep Methods Lecture 11 Alessandra Nardi Thanks to Prof. Jacob White, Deepak Ramaswamy, Michal Rewienski, and Karen Veroy

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

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

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:

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

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

Multistep Methods – Convergence Analysis Two types of error

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.

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.

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

One-step Methods – Convergence Analysis Consistency for Forward Euler Proves the theorem if derivatives of x are bounded

One-step Methods – Convergence Analysis Convergence Analysis for Forward Euler

One-step Methods – Convergence Analysis Convergence Analysis for Forward Euler

One-step Methods – Convergence Analysis A helpful bound on difference equations

One-step Methods – Convergence Analysis A helpful bound on difference equations

One-step Methods – Convergence Analysis Back to Convergence Analysis for Forward Euler

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).

Multistep Methods Definition and Observations Multistep Equation: How does one pick good coefficients? Want the highest accuracy

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

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

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

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.

Multistep Methods – Making LTE small Exactness Constraints If As any smooth v(t) has a locally accurate Taylor series in t: if Then

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

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

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

Multistep Methods – Making LTE small 10 -4 -3 -2 -1 -15 -10 -5 FE LTE Trap Best Explicit Method has highest one-step accurate Beste Timestep

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

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

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?

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

An Aside on Solving Difference Equations To understand how h is derived, first a simple case

An Aside on Solving Difference Equations Three important observations

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

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.

Multistep Methods – Stability Stability Theorem Proof Im Re -1 1

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)

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

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.

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.

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:

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