Initial-Value Problems for ODEs. numerical errors (round-off and truncation errors) Consider a perturbed system: dz dt

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1 Ital-Value Problems or ODEs d GIVEN: t t,, a FIND: t or atb umercal errors (roud-o ad trucato errors) Cosder a perturbed sstem: dz t, z t, at b z a a Does z(t) (t)? () (uqueess) a uque soluto (t) exsts a such that t C a b t z t () (well-posed) or a >, wheever ad, wth, a uque solto to the problem dz t, z t, at b z a a exsts wth z t t, at b d GIVEN: t t,, a FIND: t or atb sucet codtos or the problem to be well posed: a d GIVEN: t t,, a FIND: t or at b Dscretzato: provde t a t, t, L, (Lpschtz codto); C a, b ad satses L or some L GIVEN:,,,, FIND: 3 4

2 Classcato: Talor methods (explct, oe-step method) Explct Schemes: F,,,, Implct Schemes: F,,,,, oe-step method: F or F, mult-step method: F m,, or F,,, m Talor method o order : GIVEN: ad t t, FIND: t t t t t t O! 5 6 eed compute t, t, t,, t t, t t d d t t tt tt, d t t d tt tt, otce that t, t, t d d d t t t t t,, 7 d, t t t t t t t t t 8

3 Talor method o order : t t t t! trucato error per tme step (rom t to t ) O accumulated trucato error rom t to t T : TO OT orward Euler method ( =, explct, oe-step): t, accumulated trucato error ~ O(T ) smple but poor accurac.e. t t, orward derece hgher order (hgher ): better accurac but too much trouble seldom use 9 bacward Euler method (mplct, oe-step): t t, bacward derece d GIVEN: t t,, a FIND: t or at b t exact: t tt, t a t, accumulated trucato error ~ O(T ), t, t

4 Cra-Nlcoso method (trapezodal, mplct, oe-step): t, t, accumulated trucato error O T leap-rog method (explct, two-step): t, t t t t accumulated trucato error O T.e. t t, cetral derece 3 Ruge-Kutta methods (explct, oe-step) d order method: accumulated trucato error ~ O( T ) t, t, What values o,,, are requred to obta a accurac o order? O t O t P.S. O O t t 3 4 O t 3 O t O the other had, the Talor seres expaso o + about t s: d O 3 3 O t Thus, to have a scheme wth a trucato error/step ~ O( 3 ), we eed 5 d Ruge-Kutta method: t, t, 3 trucato error 4 3 t 6 t 6

5 d Ruge-Kutta method: (),, (mdpot method) t, (), (mdpot Euler method) t, (3) 4, 3 4, 3 (Heu's method),, t t 3 t, rd Ruge-Kutta method: t, t, 3 3 t, parameters! 6 costrats or O( 4 ) per tme step! 7 8 4th Ruge-Kutta method (the most commo oe): t, t, 3 t, t, O 6 5 accumulated trucato error OT 4 Adams-Bashorth methods o order m (explct, mult-step) t, j j j m b O j j j m m b b b O m m e.g. Adams-Bashorth methods o order 3 (explct, mult-step): b b b -m+ -m

6 Adams-Bashorth methods o order 3 (explct, mult-step): b b b b 3 4 b O b O 3 b 4b b b b b b O 3 4 O Thereore, b bb b b b 4b 3 Cocluso: b 3 b 6 b Adams-Bashorth three-step method accumulated trucato error ~ O(T 3 ) Adams-Moulto methods o order m+ (mplct, mult-step) -m+ -m+ - + m b O j j j m m b b b b O m m e.g. Adams-Moulto methods o order m+=3 (mplct, mult-step): b b b Adams-Moulto methods o order m+=3 (mplct, mult-step): b b b b O 6 b b O b b b b b b b O O

7 predct-corrector methods Thereore, b b b b b b b 3 b 5 b 8 b Cocluso: 5 8 Adams-Moulto two-step method accumulated trucato error ~ O(T 3 ) PREDICTOR: use Adams-Bashorth m-step method to predct : m * j j m j b O t, * * accumulated trucato error ~ O(T m+ ) advatage: explct dsadvatage: eed ma more computatos * CORRECTOR: use Adams-Moulto m-step method to predct : b b O m * j j m j 5 6 Geeral mult-step methods: a a a m m b b b b m m Geeral mult-step methods: a a a m m b b b b m m 3 6 (expected) 3 6 (expected) b b explct schemes mplct schemes AB/AM methods: a, a or j j m explct schemes degrees o reedom (# o adjustable varables): m mplct schemes m explct schemes maxmum order o accurac attaable: m mplct schemes 7 8

8 Sstems o deretal equatos T Let U u, u,, u ad F,,,, The N N T du t, u, u,, un du t, u, u,, un dun N t, u, u,, un I.C.s: u =, u =,, u N = N u t, u, u,, un u t, u, u,, un d u t, u, u,, u N N N 9 du F t, U U,,, e.g. Adams-Moulto two-step method (3rd order) 5 8 u u u u 5 8 u u N N N N N N T 3 t, u, u,, u N t, u, u,, un 5 N t, u, u,, u N u t,,,, u u u u N u u t, u, u,, un 8 u N u N N t, u, u,, u N t, u, u,, un t, u, u,, un N t, u, u,, u N N N 3 N Let d m,, d m, d, d Hgher order equatos t m m m d d d I.C.s =, =, = 3,, = m m u t t du d u t du d u3 t um t m dum d m du u t du u3 t dum um t dum t, u, u,, um I.C.s: u =, u =,, u m = m 3

9 Cosstec: derece equato deretal equato as? (trucato errors as ) Stablt: computed soluto to the derece equato exact soluto to the derece equato? (roud-o errors uder cotrol) test problem the exact soluto s:, t t t C exp cos s t t t t r r where t s bouded or all t as log as. r Covergece: computed soluto to the derece equato exact soluto to the deretal equato as? Schemes stablt, t t t Numercal Stablt () orward Euler method: or e e Alteratvel, the soluto o the derece equato s eve. Suppose + e, where e s the roud-o error. The (umercal stablt stead o phscal stablt) r Th s e e. u e e 35 () orward Euler method stablt crtero: ~ a lmtato o the magtude o besdes the cosderato o accurac 36

10 () bacward Euler method: bacward Euler method z 3 e z z z or z! 3! () z z or z z () stable. Icosstet wth the exact soluto whe r ad! because the Talor's seces o has a covergece radus o. z 37 Regos I, II, ad V: trucato errors uder cotrol. Regos I, ad III: roudg errors out o cotrol. 38 Rego I: Both seres coverge to a value > Roudg error dverges. Rego II: Exact seres coverge to a value > Numercal seres coverge to a value <. Roudg error coverges. 39 4

11 Rego III: Exact seres coverges to a value >. Numercal seres does ot coverge. Roudg error dverges. Rego IV: Exact seres coverges to a value >. Numercal seres does ot coverge. Roudg error coverges. 4 4 Rego V: Both seres coverge to a value <. Roudg error coverges. Rego VI: Exact seres coverges to a value <. Numercal seres does ot coverge. Roudg error coverges

12 (3) Trapezodal (Cra-Nlcoso) method: stable r (4) leaprog method: Suppose. Substtute to the derece equato to ota, C C stable, However, sce stable, Wrte exp. The exp. Moreover, s 45 stable ad r 46 (5) Ruge-Kutta methods: d method: + 3 3rd method: d method: d order Ruge-Kutta methods:, t stable 48

13 I geeral: a a a m m b b b b m m p z z a z a z a Dee q z b z b z b m m m m m m m The mult-step method s sad to be stable all roos o p(z) le the ds z ad each root o modulus s smple. The method s sad to be cosstet p ad p q Cocluso: mplct schemes: more stable allow a larger tme cremet troublesome explct schemes: less stable eed a small eas to mplemet. Theorem For the mult-step method to be coverget, t s ecessar ad sucet that t be stable ad cosstet sstem o equatos: test problem: du AU where A s a costat complex matrx. N suppose are the complex egevalues o the matrx A Boudar-Value Problems or ODEs g x,,, a xb a, b ~ shootg method ad te derece method stable all stable rego 5 5

14 shootg method IV ODEs u x x du u dx du gx, u, u dx u a, u a t g x,,, a xb a, b STEP: guess a t ad tegrate the ODEs utl xb: b STEP: chec the relatve error? ot, re-guess a. a t 53 t STEP: How to re-guess? Notce Dee t t. b, t u b t b b. o t t root-searchg mehods. e.g., Secat method:, the STEP: tae two tal guesses a = t ad a = t. b Thus we are loog or a value o t such that t = t t t t t t t btt t t t b b t ub, tt t u b, t u b, t 54 A specal case: the ODE s lear Fte-derece methods g x,,, a xb a, b (),, b g x p x q x r x a a p x q x r x a x a p x q x () a x The b b x x x 55,, x g x x x e.g. cetral derece: g x,, h h trucato error ~ O(h ) or =,,, N BC's:, N mplct equatos or,,,3,, N root-searchg or mult-varable sstem 56

15 lear equato: g x,, px qx r x p x q x r x h h p p q r h h h h h p p q r h h h h h p p q r h h h h h pn pn q N rn N h h h h h N 57