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

Transcription

1 Introduction to Simuation - Lecture 14 Mutistep Methods II Jacob White Thans to Deepa Ramaswamy, Micha Rewiensi, and Karen Veroy

2 Outine Sma Timestep issues for Mutistep Methods Reminder about LTE minimization A nonconverging exampe Stabiity + Consistency impies convergence Investigate Large Timestep Issues Absoute Stabiity for two time-scae exampes. Osciators.

3 Noninear Differentia Equation: Basic Equations Genera Notation d xt () = f ( xt (), ut ()) dt ( ) -Step Mutistep Approach: α ˆ ˆ, ( ) x = t β f x u t xˆ 2 xˆ 1 xˆ xˆ = 0 = 0 Mutistep coefficients Soution at discrete points t t 3 t 2 t 1 t Time discretization

4 Simpified Probem for Anaysis Scaar ODE: Scaar Mutistep formua: d v () t = λ v (), t v ( 0) = v 0 λ dt α ˆ ˆ v = t β λv = 0 = 0 Must Consider ALL λ Im ( λ ) O sc Decaying Soutions i a ti Growing Soutions Re( λ ) o n s

5 Convergence Anaysis Convergence Definition Definition: A mutistep method for soving initia vaue probems on [0,T] is said to be convergent if given any initia condition max vˆ v t 0 as t 0 T 0, t ( ) vˆ computed with t t vˆ computed with 2 v exact

6 Convergence Anaysis Two Conditions for Convergence 1) Loca Condition: One step errors are sma (consistency) Typicay verified using Tayor Series 2) Goba Condition: The singe step errors do not grow too quicy (stabiity) Muti-step ( > 1) methods require carefu anaysis.

7 Mutistep formua: Convergence Anaysis Goba Error Equation α v t β λv = 0 = 0 ˆ ˆ = 0 Exact soution Amost satisfies Mutistep Formua: Goba Error: E v t v d α v t t v t = e ( ) β ( ) = 0 = 0 dt ( ) ˆ Loca Truncation Error (LTE) Difference equation reates LTE to Goba error 1 ( α λ β ) ( α λ β ) ( α λ β ) t E + t E + + t E = e

8 Maing LTE Sma Exactness Constraints Loca Truncation Error: If Can't be from d v t dt () = λv() t d v t = t v t = pt dt () p () p 1 (( ) ) = 0 = 0 p d α v t t v t = e ( ) β ( ) = 0 = 0 dt (( ) t) LTE p 1 α t t β p = v( t ) d ( ) v t dt e

9 Maing LTE Sma Exactness Constraint =2 Exampe p p 1 Exactness Cons traints: α ( ) β p( ) = 0 = 0 = 0 p=0 p=1 p=2 p=3 p=4 For =2, yieds a 5x6 system of equations for Coefficients α α Note α α i = = 0 β β β 2 Aways

10 α α β 0 = β β 2 16 Maing LTE Sma Exactness Constraint =2 exampe, generating methods First introduce a normaization, for exampe α = 1 Sove for the 2-step method with owest LTE α = 1, α = 0, α = 1, β = 1/ 3, β = 4 / 3, β = 1/ ( ) 5 Satisfies a five exactness constraints LTE = C t Sove for the 2-step expicit method with owest LTE α0 = 1, α1 = 4, α2 = 5, β0 = 0, β1 = 4, β2 = 2 Can ony satisfy four exactness constraints LTE = C t 0 ( ) 4

11 L T E Mutistep Methods d vt () = vt () d Trap Maing LTE Sma LTE Pots for the FE, Trap, and Best Expicit (BESTE). Beste FE Best Expicit Method has highest one-step accurate Timestep

12 Maing LTE Sma Goba Error for the FE, Trap, and Best Expicit (BESTE). M a x E r r o r d vt () = vt () t 0,1 d FE [ ] Trap Where s BESTE? Timestep

13 M a x E r r o r Mutistep Methods worrysome d vt () = vt () d FE Maing LTE Sma Goba Error for the FE, Trap, and Best Expicit (BESTE). Trap Beste Best Expicit Method has owest one-step error but goba errror increases as timestep decreases Timestep

14 Stabiity of the method Difference Equation Why did the best 2-step expicit method fai to Converge? Mutistep Method Difference Equation 1 ( α λ β ) ( α λ β ) ( α λ β ) t E + t E + + t E = e v t v ( ) ˆ Goba Error We made the LTE so sma, how come the Goba error is so arge? LTE

15 Stabiity of the method Stabiity Definition Mutistep Method Difference Equation 1 ( α λ β ) ( α λ β ) ( α λ β ) t E + t E + + t E = e Definition: A mutistep method is stabe if as T ( ) max E C T max e T T 0, t 0, t interva t dependent t 0 Stabiity means: Goba Error is bounded by a constant times the sum of the LTE s

16 Given a th order difference eqn with zero initia conditions 1 ax+ + ax = u, x = 0,, x = 0 0 h Aside on difference Equations x can be reated to the input u by Q M q 1 = q= 1 m= 0 Root mutipicity γ qm, () m ( ς ) Convoution Sum q Roots of Root Reation x = h u = 0 convoution sum = az az a

17 Aside on difference Equations Convoution Sum Bounding Terms If If ς ς Q M q 1 ( ) m ( ) x = γqm, ςq u q= 1 m= 0 = 0 R qm, <1, then R Cmax u q q, m ( ε) t q 0 q, Independent of e <1+, hen R C max u ε ε Bounds distinct Roots

18 Stabiity of the method Stabiity Theorem Theorem: A mutistep method is stabe if and ony if Roots of 1 α0z α1z α = 0 either: 1. Have magnitude ess than one 2. Have magnitude equa to one and are distinct

19 Stabiity of the method Stabiity Theorem Proof Given the Mutistep Method Difference Equation 1 ( α λ β ) ( α λ β ) ( α λ β ) t E + t E + + t E = e ( t ) z + ( t ) If, as t 0, roots of α λ β + α λ β = ess than one in magnitude or are distinct and bounded by 1 + κ t, κ > 0 Then from the aside on difference equations κ t κt e Ce T max E C max e max t T t T T T 0, 0, 0, t t t CT ( ) e

20 roots of α z = 0 = 0 Im Stabiity of the method Stabiity Theorem Picture As t 0, roots move inward to match α poynomia -1 1 Re ( α ) λ β roots of t z = 0 for a nonzero t = 0

21 Best expicit 2-step method Stabiity of the method The BESTE Method α = 1, α = 4, α = 5, β = 0, β = 4, β = roots of z + 4z 5 = 0 Im Re Method is Widy unstabe!

22 Stabiity of the method Dahquist s First Stabiity Barrier For a stabe, expicit -step mutistep method, the maximum number of exactness constraints that can be satisfied is ess than or equa to (note there are 2-1 coefficients). For impicit methods, the number of constraints that can be satisfied is either +2 if is even or +1 if is odd.

23 Convergence Anaysis Conditions for convergence, stabiity and consistency 1) Loca Condition: One step errors are sma (consistency) Exactness Constraints up to p 0 (p 0 must be > 0) T 0, t ( ) p max e C t for t < t 1 0 2) Goba Condition: One step errors grow sowy (stabiity) roots of α z = 0 = 0 max Convergence Resut: Inside the unit circe or on the unit circe and distinct T E C max e T 2 T 0, t 0, t t max T 0, t E CT t p ( ) 0

24 sma t Bacward-Euer Computed Soution Large timestep stabiity Two time-constant circuit Circuit Exampe d xt () = Axt () dt eig( A ) = 2.1, 0.1 arge t With Bacward-Euer it is easy to use sma timesteps for the fast dynamics and then switch to arge timesteps for the sow decay

25 Large Timestep Stabiity FE on two time-constant circuit? Forward-Euer Computed Soution The Forward-Euer is accurate for sma timesteps, but goes unstabe when the timestep is enarged

26 Scaar ODE: Forward-Euer: Bacward-Euer: Large Timestep Stabiity FE, BE and Trap on the scaar ode probem d v () t = λ v (), t v ( 0) = v 0 λ dt ( 1 λ ) 1 ˆ ˆ ˆ ˆ v + = v + tλv = + t v If 1+ tλ > 1 the soution grows even if λ<0 vˆ = vˆ + tλvˆ vˆ = vˆ If 1 the soution decays even if λ 0 1 tλ < > Trap Rue: ( ) ( 1 tλ) ( tλ ) ( tλ ) ˆ ˆ ˆ v = v tλ v + vˆ vˆ 1 = vˆ

27 Forward Euer Im(z) z Large Timestep Stabiity FE arge timestep region of absoute stabiity = 1+ tλ ( ) ODE stabiity region Im( λ ) -1 Difference Eqn Stabiity region 1 Re(z) 2 t Region of Absoute Stabiity Re( λ )

28 Im(z) Large Timestep Stabiity FE arge timestep stabiity, circuit exampe Circuit exampe with t = 0.1, λ = 2.1, 0.1 ODE stabiity region Im λ ( ) -1 Difference Eqn Stabiity region 1 Re(z) 2 t Region of Absoute Stabiity Re( λ )

29 Unstabe Difference Equation Mutistep Methods Im(z) Large Timestep Stabiity FE arge timestep stabiity, circuit exampe Circuit exampe with t=1.0, λ = 2.1, 0.1 ODE stabiity region Im λ ( ) -1 Difference Eqn Stabiity region 1 Re(z) 2 t Region of Absoute Stabiity Re( λ )

30 Large Timestep Stabiity BE arge timestep region of absoute stabiity Bacward Euer ( ) 1 Im(z) z = 1 tλ Im( λ ) -1 Difference Eqn Stabiity region 1 Re(z) Region of Absoute Stabiity

31 Large Timestep Stabiity BE arge timestep stabiity, circuit exampe Circuit exampe with t = 0.1, λ = 2.1, 0.1 Im(z) Im λ ( ) -1 Difference Eqn Stabiity region 1 Re(z) Region of Absoute Stabiity

32 Im(z) Large Timestep Stabiity BE arge timestep stabiity, circuit exampe Circuit exampe with t =1.0, λ = 2.1, 0.1 Stabe Difference Equation Im λ ( ) -1 Difference Eqn Stabiity region 1 Re(z) Region of Absoute Stabiity

33 Large Timestep Stabiity Stabiity Definitions Region of Absoute Stabiity for a Mutistep method: λ t α λ tβ z ( ) Vaues of where roots of = 0 are inside the unit circe. A-stabe: A method is A-stabe if its region of absoute stabiity incudes the entire eft-haf of the compex pane Dahquist s second Stabiity barrier: There are no A-stabe mutistep methods of convergence order greater than 2, and the trap rue is the most accurate. = 0

34 Mutistep methods 4 2 t = 0.1 Numerica Experiments Osciating Strut and Mass Forward-Euer Trap rue Bacward-Euer Why does FE resut grow, BE resut decay and the Trap rue preserve osciations

35 Forward Euer Im(z) z Large Timestep Stabiity FE arge timestep osciator exampe = 1+ tλ ( ) ODE stabiity region Im( λ ) unstabe osciating -1 Difference Eqn Stabiity region 1 Re(z) 2 t Region of Absoute Stabiity Re( λ )

36 Large Timestep Stabiity BE arge timestep osciator exampe Bacward Euer ( ) 1 Im(z) z = 1 tλ Im( λ ) decaying osciating -1 Difference Eqn Stabiity region 1 Re(z) Region of Absoute Stabiity

37 Trap Rue Im(z) z = osciating Large Timestep Stabiity Trap arge timestep osciator exampe ( tλ ) ( tλ ) Im( λ ) osciating -1 Difference Eqn Stabiity region 1 Re(z) Region of Absoute Stabiity

38 Large Timestep Issues Two Time-Constant Stabe probem (Circuit) FE: stabiity, not accuracy, imited timestep size. BE was A-stabe, any timestep coud be used. Trap Rue most accurate A-stabe m-step method Osciator Probem Forward-Euer generated an unstabe difference equation regardess of timestep size. Bacward-Euer generated a stabe (decaying) difference equation regardess of timestep size. Trapezoida rue mapped the imaginary axis

39 Summary Sma Timestep issues for Mutistep Methods Loca truncation error and Exactness. Difference equation stabiity. Stabiity + Consistency impies convergence. Investigate Large Timestep Issues Absoute Stabiity for two time-scae exampes. Osciators. Didn t ta about Runge-Kutta schemes, higher order A-stabe methods.