Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010

Transcription

1 Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid Runge-Kua Choice of ime sep Errors in Numerical Simulaion Jones and Luyen handou Thornley & Johnson, 1990, pp Keen & Spain (1992) Ch 5 handou

2 Week 2 Objecives 1) To learn simple mehods for numerically simulaing dynamic biological and agriculural sysem models ha are firs order differenial equaions 2) To undersand errors associaed wih numerical simulaion of models and wha o consider when implemening he numerical mehod

3 Coninuous Dynamic Models Play imporan role in sysems analysis Derived from undersanding of sysem Physical, biological, economic Simplificaion of realiy dx ( ) General form = f { X ( ), u ( ), p)} Example dn( ) = kn( ) wih N() = sae variable, = ime, and k = a parameer

4 Basis for Simulaing Coninuous Sae Variable Models Solve for values of all sae variables over a specified ime period or a condiion is me Saring values of sae variables are needed (iniial condiions) Discreize ime, keeping coninuous sae variables Simulaion of coninuous models using discree ime is an approximaion of he exac soluion Many mehods; hese are reaed more compleely in numerical analysis courses

5 Basis for Simulaing Coninuous Sae Variable Models Definiion of he derivaive of a sae variable wih respec o ime Derivaive dx ( ) = lim X 0 + = X Approximae soluion for X(+Δ) (i.e., one sep X + X + dx

6 Basis for Simulaing Coninuous Sae Variable Models Repea he process for each poin in ime Needed o solve he model using his numerical approach are: X - he iniial condiion Δ - he ime sep dx - he model, is parameers, and inpus Wha is X +2Δ?

7 Solving for X +2Δ X +2Δ = X +Δ + Δ. (dx /) +Δ Wha is X +2Δ?

8 Biological Growh of Cells Growh Rae Growh Rae = kn() hus, N dn( ) = kn( )

9 A Simple Example: Biological Growh of Cells Model: N() = number of cells in a dish a ime is ime in days is he cell division rae consan dn( ) = kn( ) Numerical soluion (Euler Mehod finie difference mehod) N + N + dn

10 Solving A Simple Example: Biological Growh of Cells Le = 0 iniially N = N 0 = 1 cell (iniial condiion) k = 0.2 Δ = 1 day Numerical soluion Wha if Δ = 0.5 day? dn( ) N N N or = kn( ) dn N ( k N ) (0.2 1) N = 1 = 1.2 0

11 Example Comparing Numerical wih Exac Soluion Solving, wih: N =0 = 1.0 Δ = 1.0 k= 0.2 Numerical Soluion dn N + N + Exac Soluion: N ( ) = N(0) e k Number of Cells Biological Growh, dela = k=0.2,exac ime, days k=0.2,approx

12 N Example Comparing Numerical wih + N dn + Now, solving wih: N =0 = 1.0 Δ = 0.25 k= 0.2 Exac Soluion Number of Cells Biological Growh, dela = k=0.2,exac k=0.2,approx ime, days

13 Wha if here is more han one sae variable? Raes of ALL sae variables mus be compued a ime Then, all sae variables are updaed for ime +Δ Thus, if here are wo sae variables, one would compue raes of boh X 1 and X 2 a ime, hen compue X 1,+ Δ and X 2,+Δ

14 Program Seps Are Shown in Flowchar N + = N + f ( N, ) N + wih = N + f ( N, ) dn f ( N, ) = f wih ( N, ) = dn

15 There Are Oher Mehods We have shown he Euler mehod We will look a wo addiional examples: Trapezoidal Runge-Kua All compue X +Δ and are numerical approximaions Some are more accurae Choice of mehod?

16 Trapezoidal Mehod Biological Growh, dela = 1.0 f[n(+δ)] N f es [N()] Δ +Δ 1 +2Δ 2 ime

17 Trapezoidal Mehod Biological Growh, dela = 1.0 f(n,) N f es (N,) 1.0 Δ +Δ +2Δ ime Average Rae is dashed red arrow: ½ [f (N, ) + f es (N(+Δ)]

18 Trapezoidal Mehod Thus N +Δ =N + Δ. (½. [f (N() + f es [N(+Δ)]) Or N + es dn dn+ = N + + 2

19 Trapezoidal Mehod Bu, wha is f es [N(+ Δ)] Firs, use he Euler mehod o compue N es for ime +Δ es dn N+ = N + Then, esimae he rae of change of N a ime +Δ. For he biological growh model, his is: dn es + es = k N+

20 Trapezoidal Mehod Then, you have all of he informaion needed o esimae N a ime +Δ N + es dn dn+ = N + + 2

21 Discussion Wha if you have a complex model wih many parameers and erms for f(x(),u(), p)? Wha if you have more sae variables, such as one wih wo sae variables, X 1 () and X 2 ()? f 1 (X 1 (),X 2 (),u(), p) f 2 (X 1 (),X 2 (),u(), p) How would Euler differ from he Trapezoidal mehod?

22 Fourh Order Runge-Kua Mehod

23 Esimaion of Numerical Errors Due o: Discree Δ (runcaion errors) Round-off errors caused by compuer precision and sorage of exac coninuous variable values

24 Taylor Series Expansion of a funcion near a poin x dx ( ) + 2 d x ( ) + 6 d x ( ) + 24 d x = x Noe ha he Euler Mehod is exacly equal o he firs wo erms on he rhs of his equaion Thus, if we use Euler o esimae he one-sep calculaion of x +Δ hen we effecively runcae all of he remaining erms error Euler ( ) 2 d x ( ) + 6 d x ( ) + 24 d x =

25 Esimaing approximae runcaion error If Δ is very small, hen he error in one sep (Local runcaion error) of he numerical simulaion can be approximaed by he firs erm of he error equaion. For he Euler mehod, he Local runcaion is approximaed by: Local error Euler ( ) 2 For a coninuous funcion, here is a maximum value of he second derivaive over one sep. Wihou compuing his value, we can assume ha he value is M, so 2 d x 2 2 Local error Euler ( ) 2 2 M

26 Esimaing approximae runcaion error If we solve he model over a number of local ime seps o produce he approximaion o he exac coninuous funcion over a ime inerval of {0, T}, hen we know he number of seps (N s =T/Δ), Muliplying N s by he local runcaion error, we approximae Global error: Global error = N s (T/Δ) imes Local error, or Global error or Global error Euler Euler ( ) 2 T And Global error is hus proporional o (Δ) 2 M ( ) M T 2,

27 Truncaion Error for Oher Mehods The Euler Mehod is hus said o be a firs order mehod Trapezoidal Mehod is a second order mehod; his can be shown using he Taylor Series o compare wih i And Global error is hus proporional o (Δ) 2 The Fourh order Runge-Kua mehod includes he erms in he Taylor Series equaion such ha when his derivaion is developed for his mehod, he Global Error is proporional o (Δ) 4 We do no aemp o compue his value; raher i is useful o consider when selecing (Δ)

28 Effec of Δ on Numerical Errors for differen mehods Thornley and Johnson, 2000

29 Errors Thornley and Johnson, 2000

30 Consideraions for Selecing Mehod Accuracy of numerical approximaion? Compuer ime required? Difficulies in programming and debugging? Characerisics of model? Characerisics of environmenal daa, u()? Sabiliy of model equaions (i.e., anicipaion of adding more model componens?

31 Difference Models Ofen referred o as discree ime models Generally, same as Euler inegraion of a coninuous differenial equaion model Time sep (Δ) is implici in he model form Example biological growh model wrien as a difference model: N +1 = N + kn This model shows an implici Δ of 1 ime sep, as noed by he subscrip (+1).

32 Consideraions for Selecing Δ How fas he sysem changes is a major consideraion Rapidly changing sysems need a smaller Δ han sysems changing more slowly One way o deermine how rapidly a sysem changes is o look a parameers ha are producs of sae variables, such as k in he biological model Some sysems achieve a seady sae when parameers and environmenal inpus are consan and a ime consan, τ, which can be compued from k Rule of Thumb is o se Δ < τ/10

33 Consideraions for Selecing Δ In hose sysems, τ = (1/k) τ is he ime he sysem reaches 67% of is final response Example: ime consan = 2.5 (k=0.4) dela = 0.25 dx ( ) = k ( X ( ) X E ) for k=0.4, X0 = 10, XE = exac numerical

34 Consideraions for Selecing Δ In hose sysems, τ = (1/k) τ is he ime he sysem reaches 67% of is final response Example: ime consan = 2.5 (k=0.4) ime sep = 2.0 dx ( ) = k ( X ( ) X E ) for k=0.4, X0 = 10, XE = 50 exac numerical

35 Wha happened here? ime consan = 2.5 (k=0.4) exac numerical

36 Consideraions for Selecing Δ How fas he sysem changes is a major consideraion Rapidly changing sysems need a smaller Δ han sysems changing more slowly Anoher approach is o ry differen Δ values. Thus, one could decrease Δ from an iniial guess unil resuls are affeced very lile (sufficienly small changes sae variables)

37 Consideraions for Selecing Δ Accuracy? Compuer ime required? Ease of programming? Sabiliy of numerical soluion? Characerisics of model? Characerisics of environmenal daa, u()? Sabiliy of model equaions (i.e., anicipaion of adding more model componens?

38 Model Seady Sae See Keen and Spain maerial Some sysems may reach seady sae, bu no all will Sae Variables remain consan as ime increases Thus, rae of change, dx/, is zero afer some poin in ime Deermining if a sysem has seady sae: Simulae for long ime period and see if all variables reach a consan value as ime increases furher Se all firs order d.e. o zero and solve for all sae variables (simulaneously)

39 Programming Dynamic Models in EXCEL

40 Homework Se No. 1 Due January Draw Forresor diagrams for : The Michaelis-Menen model of enzyme kineics, p. 31, and he sysem described in Problem 2-2 in Keen and Spain, he predaor-prey ineracion sysem (p.37). 2. Using he model for Biological Growh, simulae he growh using he Euler mehod and he Trapezoidal mehods. Find he larges Δ ha will give he same accuracy (rounded o 5 decimal places) for he Euler mehod as you ge when you use a Δ of 0.05 for he Trapezoidal mehod, evaluaed afer 20 ime unis (i.e., simulae from ime = 0 unil ime = 20 unis). Also simulae growh using he exac soluion o he biological growh model (exponenial equaion.) 3. For he ank problem from Jones and Luyen, esimae seady sae values for H 1 and H 2 (and for V 1 and V 2 ) in wo ways: 1)using mahemaical relaionships, and 2) using numerical simulaion. Compue hese seady sae values for he parameer values given below. Wha are he seady sae values of H 1 and H 2 and for V 1 and V 2? [consans and parameers are: g = 9.8 m s -2, A 1 =10 m 2, A 2 =30 m 2, C 1 =0.8 m 2, C 2 =0.4 m 2, and i()=2 m 3 s -1 ]

41 Quesions?