lec14 (1)

Lec14(1) - Introduction to Simulation Lecture 14 Multistep Methods II Jacob White Thanks to Deepak Ramaswamy Michal Rewienski and Karen Veroy

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Introduction to Simulation - Lecture 14 Multistep Methods II Jacob White Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy

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Outline Small Timestep issues for Multistep Methods Reminder about LTE minimization A nonconverging example Stability + Consistency implies convergence Investigate Large Timestep Issues Absolute Stability for two time-scale examples. Oscillators.
Multistep Methods General Notation Basic Equations () ( () , () ) d xt f xt ut dt = Nonlinear Differential Equation: k-Step Multistep Approach: 00 ˆˆ , kk lj jj l j xtf x u t αβ −− == =∆ ∑∑ Solution at discrete points Time discretization Multistep coefficients 2 ˆ l x l t 1 l t 2 l t 3 l t lk t ˆ l x 1 ˆ l x ˆ x

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Multistep Methods Simplified Problem for Analysis () ( ) 0 () , 0 d vt vt v v dt λλ == ^ Must Consid ALL er Scalar ODE: 00 ˆˆ kk lj jj αβλ −− =∆ ∑∑ Scalar Multistep formula: λ∈ ^ Growing Solutions Decaying Solutions O s c i l l a t i o n s ( ) Im λ ( ) Re
Multistep Methods Convergence Analysis Convergence Definition Definition: A multistep method for solving initial value problems on [0,T] is said to be convergent if given any initial condition ( ) 0, ˆ max 0 as t 0 l T l t vv l t ⎡⎤ ⎢⎥ ⎣⎦ ∆→ exact v t ˆ computed with 2 l v ˆ computed with t l v

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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) Multi-step (k > 1) methods require careful analysis.
Multistep Methods Convergence Analysis Global Error Equation 00 ˆˆ 0 kk lj jj vt v αβλ −− == ∆= ∑∑ Multistep formula: Exact solution Almost satisfies Multistep Formula: () l jl j j l j d vt t e dt αβ −∆ = Local Truncation Error (LTE) ( ) ˆ ll l E v Global Error: Difference equation relates LTE to Global error ( ) ( ) 1 00 11 l k l tE tE e αλ β α λ + + + = "

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Multistep Methods Making LTE Small Exactness Constraints Local Truncation Error: LTE () 1 If pp d vt t p t dt =⇒ = () () Can't be from d dt λ = 00 kk l jl j j l j jj d t e dt αβ −− == −∆ = ∑∑ 1 kj k kjt t pk d t d e t j −∆ − = ±²³²´ ±²²³²²´