Numerical Methods for Ordinary Differential Equations

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1 Numerical Methods for Ordiary Differetial Equatios Braislav K. Nikolić Departmet of Physics ad Astroomy, Uiversity of Delaware, U.S.A. PHYS 460/660: Computatioal Methods of Physics

2 Ordiary Differetial Equatios d d d = F y, y ( t ), y ( t ),..., y ( t dt dt dt ) 0 Ordiary: oly oe idepedet variable Differetial: ukow fuctios eter ito the equatio through its derivatives Order: highest derivative i F Degree: expoet of the highest derivative 3 d y t y t = Example: ( ) ( ) 0 dt

3 What is Solutio of ODE? y = y( t) A problem ivolvig ODE is ot completely specified by its equatio ODE has to be supplemeted with boudary coditios: y Iitial value problem: is give at some startig value, i ad it is desired to fid y at some fial poits tf or at some discrete list of poits (for example, at tabulated itervals). Two poit bodary value problem: Boudary coditios are specified at more tha oe t ; typically some of the coditios will be specified at t i ad some at t. f t

4 What is Numerical Solutio to the Iitial Value Problem? dy( t) dt (, ( )); ( ) 0 0 = f t y t y t = y A umerical solutio to this problem geerates sequece of values for the idepedet variable t, t,, t ad a correspodig sequece of values of the depedet variable y y, y,, y so that each approximates solutio at : y( t ) y, = 0,, t

5 Euler Metod All fiite differece methods start from the same coceptual idea: Add small icremets to your fuctio correspodig to derivatives (right-had side of the equatios) multiplied by the stepsize. Euler method is a implemetatio of this idea i the simplest ad most direct form. y t y E u ler y tru e t Sigle-Step Forward Propagatio

6 Euler Algorithm for First-Order ODE dy f ( t, y) dt = y = f ( t, y) t iitialize t, y y( t ) do while i y = y + f ( t, y) t i + i i i+ ed do i t = t + t

7 Step Size Effects i Radioactive Decay dn N Aalytics: U = U N = N ( t = 0) e U U dt τ Numerics(Euler): dn U NU ( t) = NU (0) + t + O ( t) dt Ni Ni + Ni t τ ( ) Numerical solutio will deped o the step size t t τ

8 Stability of Euler Algorithm Step size if ofte limited by the stability criterio: dy at = ay y(0) =, y = e dt After Euler steps of size t: y = y ay t y = ( a t) + Approximate solutio will decay mootoically oly if is small eough: t t For a sigle decayig expoetial-like solutio (i.e. if there is oly oe max first order equatio) the existece of a stability criterio is ot a problem because has to be small for the reasos of accuracy. t a t

9 Accuracy: Discretizatio ad Roudoff Errors Local: C p+ du Number of steps for roudoff error to be comparable with the discretizatio error: N L = f ( u, t) Lε dt LE = y+ u+ ( t+ ) N u( t) = y t N f = f ( t) y( t) = f ( τ ) dτ h f ( t ) t0 = 0 Global: t+ LE = h f ( t ) f ( τ ) dτ GE = y y( t ) Method is of order iff: LE = O( h ) LE Ch h = t t t Itegrate over iterval: Lε = + h p L t f t0 Full Error: Ch N N GE = h f ( t ) f ( τ ) dτ = 0 GE = N = 0 t t LE t 0

10 Global Discretizatio Error Example Suppose we wat to fid the solutio over the iterval Divide the iterval ito equal steps so that t = T at T y( T) = e, y = a 3 ( at ) ( at) y( T) = at + +! 3! 3 ( ) ( at ) ( )( ) ( at) y = at + + 3! 3! ( at) 3 ( at) a t y( T) y = + + O ate! 3! at [ 0,T ] This is a measure of the global trucatio error, i.e., the error over a fixed rage i t. It is proportioal to the first power of the step size ad hece the Euler method is a first order method (do ot cofuse this with the fact that we are applyig it i this case to a first order equatio).

11 Reducig Higher Order ODE to System of First Order ODE Solve higher order ODEs by splittig them ito sets of first order equatios: d y dy + p( t) + q( t) y = g ( t) dt dt dz = g ( t) p( t) z q( t) y dy dt z = dt dy = z dt There is o uique way to do this: z dz dp( t) g ( t) q( t) y dy = + dt dt = + p( t) y dt dy = z p( t) y dt

12 Example: Realistic Motio of Baseball v v + ω r v ω r iitialize t, y( t ) do while i y = y + f ( t, y ) t i+ i i i t = t + t i+ ed do i F d r a g ω + d r v m = mg B v + S0v ω dt v x = x + v t x i i i x x B x vi+ = vi vvi t m y = y + v t y i+ i i v = v g t y i+ y i z = z + v t z i + i i z z S 0v xω vi+ = vi t m

13 More Realistic Modelig of Air Flow

14 ODE of Liear Harmoic Oscillator d θ g + siθ = 0 dt l for small θ siθ θ d θ g g + θ = 0, Ω = dt l l dθ Etotal = ml + mglθ dt must be coserved!

15 Euler Method for Liear Harmoic Oscillator Switch to dimesioless quatities: d θ + 0 θ = θ = θ 0 si( Ω t + φ ) dt E total dθ = + θ dt Euler discretizatio algorithm: ω+ = ω θ t E = + θ + = θ + ω t t E t t total = E + t + = + ( ω ) + θ + total ( )

16 Euler Fails o θ ( t)

17 Euler Fails o ω( t)

18 Euler Fails o Phase Space Trajectory

19 Ca We Resurrect Euler by Usig Smaller Step Size?

20 Cromer Fixed Euler Method for LHO = t ω ω θ + ω ω θ θ ω = + t = t + t Apparetly trivial trick, but: E + = E + t + O t θ = θ si( t t ), ω = θ cos( t t ) ( ) ( 3 ω θ ) t t 0 ω θ = θ cos( ) = 0 over a period t

21 From Euler to Higher Order Algorithms y = y + f ( t, y ) + t = t + h ( ) ( ) + vs. Mea value theorem exact y t + t = y t + dy dt t t m

22 Midpoit Method: Secod Order Ruge-Kutta s = f ( t, y ) h h s = f ( t +, y + s ) y = y + hs + O( h ) + t = t + h + 3

23 Classical Ruge-Kutta Fourth-order method s = f ( t, y ) h h s = f ( t +, y + s ) h h s3 = f ( t +, y + s ) s = f ( t + h, y + hs ) 4 3 h y = y + ( s + s + s + s ) + O( h 5 ) t = t + h +

24 Classical Ruge-Kutta F90 Subroutie

25 Geeral Sigle-Step Methods Each of the k stages of the algorithm computes slope by evaluatig (, ) for a particular value of ad a value of obtaied by takig liear combiatios of the previous slopes: f t y t y i s = f ( t + α h, y + h β s ), i =,, k i i i, j j j= The proposed step is also a liear combiatio of the slopes: k y = y + h γ s + i i i = Error is estimated from yet aother liear combiatio of the slopes: e = h δ s + i i i = k The parameters are determied by matchig terms i the Taylor series expasio of the slopes the order of the method is the expoet of the smallest power of h that caot be matched. I MATLAB ODE umerical routies are amed as odexx, where idicates the order ad xx is some special feature of the method. s i

26 Example: MATLAB ode3 Fuctio (Bogacki ad Shampie BS3 Algorithm) s = f ( t, y ) h y = y + (s + 3s + 4 s ) h h s = f ( t +, y + s ) 3 3 s3 = f ( t + h, y + hs ) 4 4 h e = ( 5s + 6s + + 8s 9 s ) t = t + h; s = f ( t, y ) 7 3 4

27 Beyod Ruge-Kutta Methods Ruge-Kutta methods propagates a solutio over a iterval by combiig the iformatio from several Euler-style steps (each ivolvig oe evaluatio of the right-had side f s), ad the usig the iformatio obtaied to match Taylor series expasio up to some higher order. Richardso extrapolatio method used the powerful idea of extrapolatig computed result to the value that would have bee obtaied if the stepsize had bee very much smaller tha it actually was. I particular, extrapolatio to zero stepsize is the desired goal implemeted by Burlich-Stoer algorithm. Predictor-corrector methods store the solutio alog the way, ad use those results to extrapolate the solutio oe step advaced; they correct the extrapolatio usig derivative iformatio at the ew poit.

28 Stiff Systems of Differetial Equatios Stiffess arises i systems of ODE where there are two or more very differet scales of the idepedet variable: du dv = 998u v, = 999u 999v u = y z u = e e dt dt v = y + t z u(0) =, v(0) = 0 v = e + e Follow the variatio i the solutio o the shortest legth scale to maitai stability of the itegratio eve though accuracy requiremets allow for a much larger step size use implicit methods: explicit y = cy, c > 0 y = y + ty = ( c t ) y + t > c y as implicit = + = = + c t y cy y y ty y y 000 t t 000t

29 Solutios to Stiffess Beyod Implicit Euler To improve higher-order (tha Euler, which is first-order) methods use: Geeralizatios of Ruge-Kutta methods Rosebrock methods ad Kaps-Retrop methods. Burlich-Stoer algorithm geeralized to Bader-Deuflhard semi-implicit extrapolatio method. Predictor-corrector methods geeralized to Gear backward differetiatio method.

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