Introduction to Simulation - Lecture 13. Convergence of Multistep Methods. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy

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1 Introduction to Simuation - Lecture 13 Convergence of Mutistep Methods Jacob White Thans to Deepa Ramaswamy, Micha Rewiensi, and Karen Veroy

2 Outine Sma Timestep issues for Mutistep Methods Loca truncation error Seecting coefficients. Nonconverging methods. Stabiity + Consistency impies convergence Next Time Investigate Large Timestep Issues Absoute Stabiity for two time-scae exampes. Osciators.

3 Mutistep Methods 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 = = Mutistep coefficients Soution at discrete points t t 3 t 2 t 1 t Time discretization

4 Mutistep Methods Basic Equations Common Agorithms Mutistep Equation: Forward-Euer Approximation: α ˆ ˆ x = t β f x, u t = = ( ( )) (, ) x t x t + t f x t u t ( ) ( ) ( ) ( ) ( ) 1 xˆ xˆ = t f xˆ, u t = 1, α = 1, α = 1, β =, β = FE Discrete Equation: ( ) Mutistep Coefficients: BE Discrete Equation: 1 1 ( ) 1 xˆ xˆ = t f xˆ, u( t ) = 1, α = 1, α = 1, β = 1, β = Mutistep Coefficients: 1 1 t ˆ ˆ = ˆ + ˆ = 1, α = 1, α1 = 1, β =, β1 = 2 2 Trap Discrete Equation: ( ) Mutistep Coefficients: (, ) (, ( 1 )) ( ) 1 1 x x f x u t f x u t

5 Mutistep Methods Basic Equations Definitions and Observations Mutistep Equation: α ˆ ˆ x = t β f x, u t = = ( ( )) 1) If β the mutistep method is impicit 2) A step mutistep method uses previous x' s and f ' s 3) A normaization is needed, α = 1 is common 4) A -step method has 2+ 1 free coefficients How does one pic good coefficients? Want the highest accuracy

6 Mutistep Methods Simpified Probem for Anaysis Scaar ODE: d v () t = λ v (), t v ( ) = v λ dt Why such a simpe Test Probem? Noninear Anaysis has many unreveaing subteties Scaar is equivaent to vector for mutistep methods. d x () t = Ax () t dt Let Ey() t = x() t Decouped Equations mutistep discretization α ˆ ˆ x = t β Ax = = 1 α ˆ ˆ y = t β E AEy = = λ 1 α ˆ ˆ y = t β y = = λ n

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

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

9 Mutistep Methods Convergence Anaysis Order-p convergence Definition: A muti-step method for soving initia vaue probems on [,T] is said to be order p convergent if given any λ and any initia condition max T, t for a t ess than a given ( ) ( ) vˆ v t C t t p Forward- and Bacward-Euer are order 1 convergent Trapezoida Rue is order 2 convergent

10 M ax Muti-step Methods Bacward-Euer Convergence Anaysis Reaction Equation Exampe 1-2 E rr 1-4 Trap rue Forward-Euer o r Timestep For FE and BE, Error t For Trap, Error ( t) 2

11 Mutistep Methods 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) A one-step (=1) methods are stabe in this sense. Muti-step ( > 1) methods require carefu anaysis.

12 Mutistep Methods Mutistep formua: Convergence Anaysis Goba Error Equation α v t β λv = = ˆ ˆ = Exact soution Amost satisfies Mutistep Formua: Goba Error: E v t v d α v t t v t = e ( ) ˆ ( ) β ( ) = = dt 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 1 1

13 Forward-Euer Convergence Anaysis Consistency for Forward Euer Forward-Euer definition v vˆ + 1 vˆ tλvˆ =,( 1) (( 1) ) ( ) ( ) Substituting the exact vt and expanding τ t + t 2 2 dv( t) ( t) d v( τ ) + t v t t = 2 d dt 2 dt v = λ v dt e e where is the LTE and is bounded by 2 ( ) ( ) 2 dvτ e C t, where C =.5 max τ [, T ] 2 dt

14 Forward-Euer Forward-Euer definition vˆ + 1 = vˆ + tλvˆ Using the LTE definition Convergence Anaysis Goba Error Equation (( 1) ) ( ) ( ) v + t = v t + tλv t + e Subtracting yieds goba error equation E + 1 = I + tλ E + e ( ) Using magnitudes and the bound on e ( 1 ) ( ) E I + tλ E + e + t λ E + C t

15 Forward-Euer Convergence Anaysis A hepfu bound on difference equations A emma bounding difference equation soutions Then u ( ) + 1 If u 1 + ε u + b, u =, ε > e ε ε b To prove, first write u as a power series and sum 1 1 ( 1+ ε ) ( 1 ) u + ε b = b 1 ( 1 ε ) = +

16 One-step Methods Convergence Anaysis A hepfu bound on difference equations cont. ε To finish, note (1 + ε) e (1 + ε) e ε ( ε) ( ) ( ε) e u b = b b 1 1+ ε ε ε ε

17 One-step Methods Convergence Anaysis Bac to Forward Euer Convergence anaysis. Appying the emma and canceing terms E t λ E + C ( t )2 ε b Finay noting that t T, max [, L] C E e λ T t λ e t tλ C λ ( t) 2

18 Forward-Euer Convergence Anaysis Observations about the forward-euer anaysis. max C E e λ T t [, L] forward-euer is order 1 convergent Bound grows exponentiay with time interva. C reated to exact soution s second derivative. The bound grows exponentiay with time. λ

19 Forward-Euer 12 Convergence Anaysis Exact and forward-euer(fe) Pots for Unstabe Reaction TempExact Rexact RFE TFE Forward-Euer Errors appear to grow with time

20 Forward-Euer 1.2 E r r o r Rexact-RFE Convergence Anaysis forward-euer errors for soving reaction equation Time Texact - TFE Note error grows exponentiay with time, as bound predicts

21 Forward-Euer 1 Convergence Anaysis Exact and forward-euer(fe) Pots for Circuit..8 v1fe.6 v1exact.4 v2fe.2 v2exact Forward-Euer Errors don t aways grow with time

22 E r r o r Forward-Euer v1exact - v1fe Convergence Anaysis forward-euer errors for soving circuit equation. v2exact-v2fe Time Error does not aways grow exponentiay with time! Bound is conservative

23 Mutistep Methods 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 (( ) ) = = p d α v t t v t = e ( ) β ( ) = = dt (( ) t) LTE p 1 α t t β p = v( t ) d ( ) v t dt e

24 Mutistep Methods If if α (( ) ) β ( ) = = p Maing LTE Sma Exactness Constraints Cont. ( ) p 1 t t p t = p p p 1 t α β p = e = = ( ) ( ) ( ) p p 1 α ( ( ) ) β p( ) = then e = for v( t) = t = = α As any smooth v(t) has a ocay accurate Tayor series in t: ( ) β p( ) = for a p p = = Then p p 1 α v t d v t e C t ( ) ( ) p ( ) + 1 β = = = = dt p

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

26 Mutistep Methods Exactness Constraints for =2 Maing LTE Sma Exactness Constraint =2 Exampe Continued α α α = β β β 2 Forward-Euer α = 1, α = 1, α =, β =, β = 1, β =, ( ) 2 FE satisfies p = and p = 1 but not p = 2 LTE = C t Bacward-Euer α = 1, α = 1, α =, β = 1, β =, β =, ( ) 2 BE satisfies p = and p= 1 but not p= 2 LTE = C t Trap Rue α = 1, α = 1, α =, β =.5, β =.5, β =, ( ) 3 Trap satisfies p =,1,or 2 but not p = 3 LTE = C t

27 Mutistep Methods 1 1 α α β = 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 =, 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 α = 1, α1 = 4, α2 = 5, β =, β1 = 4, β2 = 2 Can ony satisfy four exactness constraints LTE = C t ( ) 4

28 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

29 Mutistep Methods 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,1 d FE [ ] Trap Where s BESTE? Timestep

30 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

31 Mutistep Methods 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 1 1 v t v ( ) ˆ Goba Error We made the LTE so sma, how come the Goba error is so arge? LTE

32 An Aside on Soving Difference Equations Consider a genera th order difference equation = 1 ax ax ax u Which must have initia conditions 1 1 x = x, x = x,, x = x As is cear when the equation is in update form ( + ) 1 x = a x + + a x u a Most important difference equation resut x can be reated to u by x = h u =

33 An Aside on Difference Equations Cont = 1 If az az a has distinct roots ς, ς,, ς 1 2 Then = = 1 ( ) x = h u where h = γ ς To understand how h is derived, first a simpe case Supp ose x = ς x 1 + u and x = x = ςx + u = u, x = ςx + u = ςu + u x = ς u =

34 An Aside on Difference Equations Cont. Three important observations If ς i <1 for a i, then x Cmx a u where C does not depend on If ς >1 for any i, then there exists i a bounded u such th at x If ς 1 for a i, and if ς =1, ς i i i then x Cmax u is distinct

35 Mutistep Methods Stabiity of the method Difference Equation Mutistep Method Difference Equation 1 ( α λ β ) ( α λ β ) ( α λ β ) t E + t E + + t E = e 1 1 Definition: A mutistep method is stabe if and ony if t max E C max e for a ny e T T, t, t t As T Theorem: A mutistep method is stabe if and ony if 1 The roots of α z + α z + + α = are either 1 Less than one in magnitude or equa to one and distinct

36 Mutistep Methods 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 1 1 α z = are either = If the roots of ess than one in magnitude equa to one in magnitude but distinct Then from the aside on difference equations E Cmax e From which stabiity easiy foows.

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

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

39 Mutistep Methods 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 coefficients). For impicit methods, the number of constraints that can be satisfied is either +2 if is even or +1 if is odd.

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

41 Summary Sma Timestep issues for Mutistep Methods Loca truncation error and Exactness. Difference equation stabiity. Stabiity + Consistency impies convergence. Next time Absoute Stabiity for two time-scae exampes. Osciators. Maybe Runge-Kutta schemes