CSE 245: Computer Aided Circuit Simulation and Verification Fall 2004, Nov Lecture 8: Numerical Integration

Outline One-step Method for ODE (IVP) Linear MultiStep Method Forward Euler Backward Euler Trapezoidal Rule Equivalent Circuit Model Linear MultiStep Method Convergence Analysis Consistence Stability Time Step Control (next lecture) Stability Region Stiff System Dynamic Time Step Control Over-Relaxation Method & ADI November 27, 2018

Circuit Equation Formulation For dynamical circuits the equations can be written compactly: For sake of simplicity, we shall discuss first order ODEs in the form: November 27, 2018 courtesy Alessandra Nardi UCB

Ordinary Differential Equations Typically analytic solutions are not available  solve it numerically November 27, 2018 courtesy Alessandra Nardi UCB

Ordinary Differential Equations Assumptions and Simplifications Not necessarily a solution exists and is unique for: It turns out that, under rather mild conditions on the continuity and differentiability of F, it can be proven that there exists a unique solution. Also, for sake of simplicity only consider linear case: We shall assume that has a unique solution November 27, 2018 courtesy Alessandra Nardi UCB

Finite Difference Methods Basic Concepts First - Discretize Time Second - Represent x(t) using values at ti Approx. sol’n Exact sol’n Third - Approximate using the discrete November 27, 2018 courtesy Alessandra Nardi UCB

Forward Euler Approximation November 27, 2018 courtesy Alessandra Nardi UCB

Forward Euler Approximation November 27, 2018 courtesy Alessandra Nardi UCB

Backward Euler Approximation November 27, 2018 courtesy Alessandra Nardi UCB

Backward Euler Approximation Solve with Gaussian Elimination November 27, 2018 courtesy Alessandra Nardi UCB

Trapezoidal Rule Approximation November 27, 2018 courtesy Alessandra Nardi UCB

courtesy Alessandra Nardi UCB Trapezoidal Rule Approximation Solve with Gaussian Elimination November 27, 2018 courtesy Alessandra Nardi UCB

Numerical Integration View Trap BE FE November 27, 2018 courtesy Alessandra Nardi UCB

Equivalent Circuit Model-BE Capacitor + + + C - - - November 27, 2018

Equivalent Circuit Model-BE Inductor + + - + L - - November 27, 2018

Equivalent Circuit Model-TR Capacitor + + + C - - - November 27, 2018

Equivalent Circuit Model-TR Inductor + + - + L - - November 27, 2018

Summary of Basic Concepts Trap Rule, Forward-Euler, Backward-Euler Are all one-step methods Forward-Euler is simplest No equation solution explicit method. Boxcar approximation to integral Backward-Euler is more expensive Equation solution each step implicit method Trapezoidal Rule might be more accurate Trapezoidal approximation to integral November 27, 2018 courtesy Alessandra Nardi UCB

courtesy Alessandra Nardi UCB Outline One-step Method for ODE (IVP) Forward Euler Backward Euler Trapezoidal Rule Linear MultiStep Method Convergence Analysis Consistence Stability Stiff System and Time Step Control (next lecture) Stiff System Dynamic Time Step Control November 27, 2018 courtesy Alessandra Nardi UCB

Linear Multistep Method (LMS) Basic Equations Nonlinear Differential Equation: k-Step Multistep Approach: Multistep coefficients Solution at discrete points Time discretization November 27, 2018 courtesy Alessandra Nardi UCB

LMS: Common Algorithm TR, BE, FE are one-step methods Multistep Equation: Forward-Euler Approximation: FE Discrete Equation: Multistep Coefficients: BE Discrete Equation: Multistep Coefficients: Trap Discrete Equation: Multistep Coefficients: November 27, 2018 courtesy Alessandra Nardi UCB

Adams-Bashforth formula 0 =0 The first order Adams-Bashforth formula (forward Euler) The second order Adams-Bashforth formula November 27, 2018

Adams-Moulton formula 0 0 The first order Adams-Moulton formula (backward Euler) The second order Adams-Moulton formula (trapezoidal) November 27, 2018

Convergence Analysis 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 November 27, 2018

LMS: Convergence Analysis 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 November 27, 2018 courtesy Alessandra Nardi UCB

LMS: 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 November 27, 2018 courtesy Alessandra Nardi UCB

Convergence Analysis (1) November 27, 2018 courtesy Alessandra Nardi UCB

Convergence Analysis (2) 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. November 27, 2018 courtesy Alessandra Nardi UCB

Convergence Analysis (3) 1) Local Condition: One step errors are small (consistency) Typically verified using Taylor Series Exactness Constraints up to p0 (p0 must be > 0) 2) Global Condition: The single step errors do not grow too quickly (stability) All one-step methods are stable in this sense. November 27, 2018 courtesy Alessandra Nardi UCB

courtesy Alessandra Nardi UCB Consistency 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 November 27, 2018 courtesy Alessandra Nardi UCB

One-step Methods – Convergence Analysis Consistency for Forward Euler Forward-Euler definition Expanding in t about zero yields Proves the theorem if derivatives of x are bounded November 27, 2018 courtesy Alessandra Nardi UCB

One-step Methods – Convergence Analysis Convergence Analysis for Forward Euler Forward-Euler definition November 27, 2018 courtesy Alessandra Nardi UCB

One-step Methods – Convergence Analysis Convergence Analysis for Forward Euler Subtracting the previous slide equations Define the "Global" error Taking norms and using the bound on l e November 27, 2018 courtesy Alessandra Nardi UCB

One-step Methods – Convergence Analysis A helpful bound on difference equations A lemma bounding difference equation solutions To prove, first write as a power series and sum u l November 27, 2018 courtesy Alessandra Nardi UCB

One-step Methods – Convergence Analysis A helpful bound on difference equations Mapping the global error equation to the lemma November 27, 2018 courtesy Alessandra Nardi UCB

One-step Methods – Convergence Analysis Back to Convergence Analysis for Forward Euler Applying the lemma and canceling terms November 27, 2018 courtesy Alessandra Nardi UCB

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). November 27, 2018 courtesy Alessandra Nardi UCB

courtesy Alessandra Nardi UCB Convergence 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 November 27, 2018 courtesy Alessandra Nardi UCB

Linear Multistep Methods (LMS) Definition and Observations Multistep Equation: How does one pick good coefficients? Want the highest accuracy November 27, 2018 courtesy Alessandra Nardi UCB

Linear Multistep Methods (LMS) 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 November 27, 2018 courtesy Alessandra Nardi UCB

Linear Multistep Methods Simplified Problem for Analysis Scalar ODE: Scalar Multistep formula: Decaying Solutions Osci l lations Growing Solutions November 27, 2018 courtesy Alessandra Nardi UCB

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 November 27, 2018 courtesy Alessandra Nardi UCB

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. November 27, 2018 courtesy Alessandra Nardi UCB

Multistep Methods – Making LTE small Exactness Constraints If As any smooth v(t) has a locally accurate Taylor series in t: if Then November 27, 2018 courtesy Alessandra Nardi UCB

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 November 27, 2018 courtesy Alessandra Nardi UCB

Multistep Methods – Making LTE small Exactness Constraints – k=2 Example Exactness Constraints for k=2 November 27, 2018 courtesy Alessandra Nardi UCB

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 November 27, 2018 courtesy Alessandra Nardi UCB

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 November 27, 2018 courtesy Alessandra Nardi UCB

Multistep Methods – Making LTE small Max Error FE Where’s BESTE? Trap Timestep November 27, 2018 courtesy Alessandra Nardi UCB

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 November 27, 2018 courtesy Alessandra Nardi UCB

LMS: Stability 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? November 27, 2018 courtesy Alessandra Nardi UCB

courtesy Alessandra Nardi UCB Convergence 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 November 27, 2018 courtesy Alessandra Nardi UCB

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 November 27, 2018 courtesy Alessandra Nardi UCB

An Aside on Solving Difference Equations To understand how h is derived, first a simple case November 27, 2018 courtesy Alessandra Nardi UCB

An Aside on Solving Difference Equations Three important observations November 27, 2018 courtesy Alessandra Nardi UCB

LMS: 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: November 27, 2018 courtesy Alessandra Nardi UCB

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 November 27, 2018 courtesy Alessandra Nardi UCB

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. November 27, 2018 courtesy Alessandra Nardi UCB

Multistep Methods – Stability Stability Theorem Proof Im Re -1 1 November 27, 2018 courtesy Alessandra Nardi UCB

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) November 27, 2018 courtesy Alessandra Nardi UCB

courtesy Alessandra Nardi UCB LMS: A-Stable 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 November 27, 2018 courtesy Alessandra Nardi UCB

courtesy Alessandra Nardi UCB LMS: Stability 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. November 27, 2018 courtesy Alessandra Nardi UCB

courtesy Alessandra Nardi UCB LMS: 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. November 27, 2018 courtesy Alessandra Nardi UCB

Stabilities Froward Euler November 27, 2018

FE region of absolute stability Forward Euler ODE stability region Im(z) Difference Eqn Stability region Region of Absolute Stability Re(z) -1 1 November 27, 2018 courtesy Alessandra Nardi UCB

Stabilities Backward Euler November 27, 2018

courtesy Alessandra Nardi UCB BE region of absolute stability Backward Euler Im(z) Difference Eqn Stability region Re(z) -1 1 Region of Absolute Stability November 27, 2018 courtesy Alessandra Nardi UCB

Stabilities Trapezoidal November 27, 2018

courtesy Alessandra Nardi UCB 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 November 27, 2018 courtesy Alessandra Nardi UCB