lec13 (1)

lec13 (1) - Introduction to Simulation - Lecture 13...

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Introduction to Simulation - Lecture 13 Convergence of Multistep Methods Jacob White Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy
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Outline Small Timestep issues for Multistep Methods Local truncation error Selecting coefficients. Nonconverging methods. Stability + Consistency implies convergence Next Time Investigate Large Timestep Issues Absolute Stability for two time-scale examples. Oscillators.
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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 Common Algorithms Basic Equations () 00 ˆˆ , kk lj jj l j xtf x u t αβ −− == =∆ ∑∑ Multistep Equation: ( ) ( ) ( ) ( ) ( ) 11 1 , ll xt tf xt ut ≈+ FE Discrete Equation: ( ) ( ) 1 ˆ , l l xx t f xu t −= 01 0 1 1, 0, 1 k α β = −= = Forward-Euler Approximation: Multistep Coefficients: BE Discrete Equation: Multistep Coefficients: 0 1 0 k = ( ) 1 ˆ , l l t f x u t Trap Discrete Equation: ( ) 1 ˆ ˆ ,, 2 l l t x x fxut fx ut + 0 1 , 22 k αα = = Multistep Coefficients:
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Multistep Methods Definitions and Observations Basic Equations () 00 ˆˆ , kk lj jj l j xtf x u t αβ −− == =∆ ∑∑ Multistep Equation: 0 1) If 0 the multistep method is implicit β≠ 2) A step multistep method uses previous ' and ' x s f s 0 3) A normalization is needed, 1 is common α= 4) A -step method has 2 1 free coefficients + How does one pick good coefficients? Want the highest accuracy
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Multistep Methods Simplified Problem for Analysis • Nonlinear Analysis has many unrevealing subtleties • Scalar is equivalent to vector for multistep methods. Why such a simple Test Problem? Le t () Ey t x t = 1 00 ˆˆ kk lj jj y tE A E y αβ −− == =∆ ∑∑ 1 n yt y λ ⎡⎤ ⎢⎥ ⎣⎦ % Decoupled Equations d xt A dt = multistep discretization xtA x () ( ) 0 , 0 d vt vt v v dt λλ ^ Scalar ODE:
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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
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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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