(5.5) Multistep Methods

Transcription

1 (5.5) Mulstep Metods Consider te inial-value problem for te ordinary differenal equaon: y t f t, y, a t b, y a. Let y t be te unique soluon. In Secons 5., 5. and 5.4, one-step numerical metods: Euler Metod, Taylor Metods of Order n and Runge-Kutta Metods of Order n are studied. Tese metods compute te current step y i based on te informaon given by te previous step y i. Can earlier steps y 0, y,...,y i also be used to generate y i? Te metods studied in tis secon compute y i using te informaon on m previous steps y i, y i,..., y i m.let b N a for a given posive integer N and define t i t i a i.. Mulstep Metods: Derivaon: Recall by te Fundamental Teorem of Calculus y y t i y t dt f t, y t dt. A difference equaon can be designed based on te following equaon: y i y i f t, y t dt were yi y t i. Now consider approximang te funcon f t, y t using i points by an it degree interpolaon polynomial P i t t 0, f t 0, y 0, t, f t, y,..., t i, f t i, y i or an i t degree interpolang polynomial P i t using i points t 0, f t 0, y 0, t, f t, y,..., t i, f t i, y i,, f, y i. Let m. An m step metod for solving an inial-value problem uses te following difference equaon to compute te approximaon y i at te points y i a m y i a m y i... a 0 y i m linear combinaon of y i,y i,...,y i m b m f, y i b m f t i, y i... b 0 f m, y i m linear combinaon of m slopes for i m,m,...,n, were a 0, a,..., a m and b 0, b,..., b m are constants and te starng values y 0, y,...,y m m are specified (tese values can be computed using an one-step metod). Explicit and Implicit Metod: A mulstep metod is explicit if b m 0andisimplicit if b m 0. Te local truncaon error for a mulstep metod: i y t i a m y t i... a 0 y t i m b m f, y... b 0 f t i m, y t i m For example, an explicit -step metod:

2 y 0 y is computed by an one-step metod y i a y a 0 y 0 b f t i,y i b 0 f t i,y i for i and an implicit -step metod: y 0 y is computed by an one-step metod y i a y a 0 y 0 b f t i, y i b f t i,y i b 0 f t i,y i for i.. Adams-Basfort Mulstep Explicit Metods: a. -step Explicit Metod: At te it iteraon, points t i, f t i, y i and t i, f t i, y i areusedtoformp t : P t f t i, y t t i i f t t i t i, y t t i i i t i t i f t i, y i t t i f t i, y i t t i P t dt f t i, y i t t i f t i, y i t t i dt f t i, y i t t i f t i, y i t t i f t i, y i 4 f t i, y i 0 t i f t i, y i f t i, y i y 0, y y i y i f t i, y i f t i, y i,fori,,...,n. Local truncaon error: i y y t i f t i, y t i f t i, y t i y y t i y t i y t i y y t i y t i y t i y t i, t i in t i, y t i y t i y t i y t i y t i y t i!,weret i in t i, t i i y t i y t i y t i y t i y t i 4 y t i y t i y t i 4 y t i y t i 4 y t i 5 y t i were t i in t i,

3 i 5 y t i,weret i is in t i,. b. -step Explicit Metod: In a similar way, wen we use te interpolang polynomial P t at t i,f t i, y i, t i,f t i, y i and t i, f t i, y i we can derive te following -step explicit metod: y 0, y, y y i y i f t i, y i f t i, y i 5 f t i, y i, for i,,..., N. Te corresponding local truncaon error is: i 8 y 4 t i,weret i is in t i, t i. c. 4-step Explicit Metod: Also in a similar way, wen we use te interpolang polynomial P t at t i,f t i, y i, t i,f t i, y i, t i,f t i, y i and t i, f t i, y i we can derive te following 4-step explicit metod: y 0, y, y, y y i y i 55 4 f t i, y i 59 4 f t i, y i 7 4 f t i, y i 9 4 f t i, y i, for i,4...,n. Te corresponding Local truncaon error is: i 5 70 y 5 t i 4,weret i is between t i, t i.. Adams-Moulton Mulstep Implicit Metods: Derivaon: Implicit metods are derived by using, f, y as addional interpolaon t point in te approximaon of te integral i f t, y t dt a. -step Implicit Metod: y 0 y i y i f, y i f t i, y i Local truncaon error: P t f, y i i y t i,weret i is in t i,. t t i t i f t i, y i t t i f, y i t t i f t i, y i t

4 P t dt f t t t i, y i i f t i, y i t f, y i 0 f t i, y i 0 t i f, y i f t i, y i b. -step Implicit Metod: Local truncaon error: y 0, y y i y i 5 f, y i 8 f t i, y i f t i, y i, for i,,...,n. i 4 y 4 t i,weret i is in t i,. How to derive te local truncaon error? i y y t i 5 f t i, y 8 f t i, y t i f t i, y t i y y t i y t i y t i y t i y 4 t i 4 5 f, y 8 f t i, y t i f t i, y t i 5 y 8 y t i y t i 5 y t i y t i y t i! y 4 t i 8 y t i y t i y t i y t i! y 4 t i, were t i in t i, t i in t i, t i y t i y t i y t i y 4 t i,weret i in i y t i y t i y t i y 4 t i 4 y t i y t i y t i y 4 t i t i, 4 y 4 t i y 4 t i 4 y 4 were t i in t i, c. -step Implicit Metod: y 0, y, y 9 y i y i f t 4 i, y i 9 f t 4 i, y i 5 f t 4 i, y i f t 4 i, y i, for i,,..., N. 4 Local truncaon error:

5 i 9 70 y 5 t i 4,weret i is in t i,. d. 4-step Implicit Metod: y 0, y, y, y y i y i 5 f t 70 i, y i 4 f t 70 i, y i 4 f t 70 i, y i 0 f t 70 i, y i 9 f t 70 i, y i, for i,..., N Local truncaon error: i 0 y t i 5,weret i is in t i,. Summary: Explicit Mulstep Metods - Adams-Basfort Metods: m y i i y i f t i,y i y t i y i f t i, y i f t i, y i 5 y t i y i f t i, y i f t i, y i 5 f t i, y i 8 y 4 t i 4 y i 55 4 f t i, y i 59 4 f t i, y i 7 4 f t i, y i 9 4 f t i, y i 5 70 y 5 t i 4 Implicit Mulstep Metods - Adams-Moulton Metods: m y i i y i f, y i f t i, y i y t i y i 5 f, y i 8 f t i, y i f t i, y i 4 y 4 t i y i 9 4 f, y i 9 4 f t i, y i 5 4 f t i, y i 4 f t i, y i 9 70 y 5 t i 4 4 y i 5 70 f, y i 4 70 f t i, y i 4 70 f t i, y i 0 70 f t i, y i 70 9 f t 0 y t i 5 i, y i Note tat for Adams-Basfort and Adams-Moulton m-step metods a m 0,..., a 0 0. Before te end of tis secon, we will study Milne s Explicit 4-step Metod and Simpson s -step Metod in wic, respecvely, a 0, a 0, a 0anda 0 ; and a 0anda 0. Example Consider te inial-value problem: y t y t, 0 t, y Let a. Use te Midpoint Metod to generate esmate: y ;

6 b. generate y using Adams-Basfort -step metod; and c. generate y using Adams-Moulton -step metod. Wat is te order of te local truncaon error for y? t 0 0, 0., y y y 0 f t 0,y y y 0 f t 0, y y y f t,y f t 0, y y y f t, y f t, y true error e clear; clf y-t.^ ; (y-t.^ )-*t; [tv,yvtaylor,n] tayfun(fun,fun,0,,/,0.); [tv,yvmidpt,n] rkmid(fun,0,,/,0.); [tv,yvab,n] adambas(fun,0,,/,0.,); [tv,yvam,n] adammoul(fun,0,,/,0.,); ysol (tv ).^-0.5*exp(tv); plot(tv,yvtaylor, r-o,tv,yvmidpt, b-*,tv,yvab, m-,tv,yvam, g-x,tv,ysol, k- ) tle( dy/dt y-t^, y(0) /, t in [0,], 0., y(t) (t )^-0.5e^t ) axis([ ]) (Or MatLab program lect5_5_ex.m)

7 5.5 dy/dt=y-t +, y(0)=/, t in [0,], =0., y(t)=(t+) -0.5e t Taylor T Midpoint Metod Adams-Basfort m= Adams-Moulton m= y(t) y(t i )-y i, = Taylor T Midpoint Metod Adams-Basfort m= Adams-Moulton m= Oter Mulstep Metods: a. Milne s Explicit Metod: y i y i 8 f t i, y i 4 f t i, y i 8 f t i, y i 7

8 wic as local truncaon error i y 5 t i were t i is in t i, b. Simpson s Implicit Metod: y i y i f, y i 4 f t i, y i f t i, y i wic as local truncaon error i 90 4 y 5 t i were t i is in t i, Te derivaon for Milne s Metod: y y t i f t, y t dt y i y i 8 f t i, y i 4 f t i, y i 8 f t i, y i Te local truncaon error: i y y t i 8 f t i, y i 4 f t i, y i 8 f t i, y i 8 4 y t i y t i y t i y t i y t i y t i y t i y t i y t i! y t i y t i y t i! y t i! y t i! y 4 t i 4! y t i y 4 t i y 4 t i 4 y 5 t i 5! 5 y 4 t i 4! y 5 t i 4! y 5 t i 4! 4 y 5 t i 5! y t i 5 y t i 9 4 y t i 9 y 4 t i y 5 t i y 5 t i 4, t i is in t i,. 5. Predictor-Corrector Scemes: In an implicit metod, te current step y i is on te bot sides of te formula for compung y i.a predictor-corrector Sceme computes y i on te rigt side by an explicit metod, called it y i and ten computes y i on te left side using y i in te formula. Te obtained y i is called a predictor and y i is called a corrector. Note tat te approximaon error of te cosen explicit metod sould be in te same order as it is for te implicit metod. For example, in te -step Adams-Moulton metod, y i sould be computed by te -step Adams-Basfort metod or a second-order Runge-Kutta metod, or a second-order Taylor metod. Exercises:. Apply te -step Adams-Basfort metod to approximate te soluons of te following inial-value problems wit 0.. Compute y witout using te MatLab program. Use te Midpoint Metod to find y. () y ty y, 0 t, y 0 () y 4 t y t4, t, y () y y t, 0 t, y 0 0 8

9 (4) y y, t 4, t y 0. Derive te -step Adams-Basfort metod.. Derive te truncaon error for te -step Adams-Moulton (trapezoidal) metod. 4. Consider te populaon model y ry y y k y. Te first term on te rigt side is known as te logisc growt term and te second one represents arvesng/predaon of te species by some oter species. Te parameters r and k are called te natural growt rate of te populaon and te environmental carrying capacity, respecvely. Let r 0.4 and k 0 and te inial populaon be.44. () Use one of te nd order Runge-Kutta Metod to approximate y t wit 0.5. () Use Adams-Basfort -step metod to approximaon y t wit 0.5. () Use Adams-Moulton -step metod to approximaon y t wit 0.5. (4) Plot sets y i obtained in (), () and (). (5) Determine te eventual populaon level (as t ) reaced from te inial populaon. 5. A genec switc is a biocemical mecanism tat governs weter a parcular protein product of a cell is syntesized or not. Te following inial-value problem as been proposed as a model for a genec switc: g g t 0.4.4g.0, 0 t, g 0 0. g () Use one of te nd order Runge-Kutta Metod to approximate y t wit 0.5. () Use Adams-Basfort -step metod to approximaon y t wit 0.5. () Use Adams-Moulton -step metod to approximaon y t wit 0.5. (4) Plot sets y i obtained in (), () and (). (5) Predict te limit of g t as t (you may enlarge te interval of t to obtain more informaon).. Apply te -step Adams-Basfort metod wit te second-order Taylor metod (for compung y )to approximate te soluon of te following inial-value problems wit given. Compute y witout using te MatLab program. a. y y e t y,0 t 0.9, y 0 0, 0. b. y y y t c. y 4y t, t 5, y ln, 0. t 4 e t, t, y 0, Apply te -step Adams-Moulton metod wit te Modified Euler metod (for compung y ) and wit te -step Adams-Basfort metod (for compung y ) to approximate te soluons of te inial-value problems given in. wit 0.. Compute y witout using te MatLab program. 9