1. understand the Runge-Kutta 2nd order method for ordinary differential equations and how to use it to solve problems.
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- Brice Ball
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1 Runge-Kutt nd Order Metod or Ordnr Derentl Equtons Autr Kw Ater redng ts cpter ou sould be ble to:. understnd te Runge-Kutt nd order metod or ordnr derentl equtons nd ow to use t to solve problems. Wt s te Runge-Kutt nd order metod? Te Runge-Kutt nd order metod s numercl tecnque used to solve n ordnr derentl equton o te orm d d ( ) Onl rst order ordnr derentl equtons cn be solved b usng te Runge-Kutt nd order metod. In oter sectons we wll dscuss ow te Euler nd Runge-Kutt metods re used to solve ger order ordnr derentl equtons or coupled (smultneous) derentl equtons. How does one wrte rst order derentl equton n te bove orm? Emple Rewrte n d d 5.e d ( ) () orm. d Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge o 7
2 Soluton d d d d In ts cse Emple Rewrte n e 5.e 5.e ( ).e d d d d sn() 5 ( ) () orm. Soluton e d d d d In ts cse sn() sn() e sn() e ( ) 5 5 Runge-Kutt nd order metod Euler s metod s gven b Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge o 7
3 ( ) () were ( ) To understnd te Runge-Kutt nd order metod we need to derve Euler s metod rom te Tlor seres. d d d d! d! d ( ) ( ) ( )... ( )( ) '( )( ) ''( )( )... ()!! As ou cn see te rst two terms o te Tlor seres ( ) re Euler s metod nd ence cn be consdered to be te Runge-Kutt st order metod. Te true error n te ppromton s gven b () E t ( ) ( )!!... So wt would nd order metod ormul loo le. It would nclude one more term o te Tlor seres s ollows. (4)! Let us te generc emple o rst order ordnr derentl equton Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge o 7
4 d d e ( ) e 5 Now snce s uncton o (5) ( ) e ( ) ( ) d d ( e ) [( e ) ]( e ) ( ) 5e 9 ( e ) Te nd order ormul or te bove emple would be! ( ) ( ) ( ) ( 5 e e 9 )! However we lred see te dcult o vng to nd ( ) Wt Runge nd Kutt dd ws wrte te nd order metod s n te bove metod. ( ) (6) were ( p q ) (7) Ts orm llows one to te dvntge o te nd order metod wtout vng to. clculte Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge 4 o 7
5 So ow do we nd te unnowns p nd q. Wtout proo (see Append or proo) equtng Equton (4) nd (6) gves tree equtons. p q Snce we ve equtons nd 4 unnowns we cn ssume te vlue o one o te unnowns. Te oter tree wll ten be determned rom te tree equtons. Generll te vlue o s cosen to evlute te oter tree constnts. Te tree vlues generll used or re nd nd re nown s Heun s Metod te mdpont metod nd Rlston s metod respectvel. Heun s Metod Here s cosen gvng p q resultng n (8) were (9) Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge 5 o 7
6 (9b) ( ) Ts metod s grpcll eplned n Fgure. Slope ( ) Slope predcted Averge Slope [ ( ) ( )] Fgure Runge-Kutt nd order metod (Heun s metod). Mdpont Metod Here s cosen gvng p q resultng n () Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge 6 o 7
7 were () (b) Rlston s Metod Here s cosen gvng p 4 q 4 resultng n () were () (b) 4 4 Emple Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge 7 o 7
8 A bll t K s llowed to cool down n r t n mbent temperture o K. Assumng et s lost onl due to rdton te derentl equton or te temperture o te bll s gven b dθ ( θ 8 dt ) were θ s n K nd t n seconds. Fnd te temperture t t 48 seconds usng Runge-Kutt nd order metod. Assume step sze o 4 seconds. Soluton dθ dt 4 8 ( θ 8 ) ( t θ).67 ( θ 8 ) Per Heun s metod gven b Equtons (8) nd (9) θ θ ( θ ) t ( t ) θ t θ θ () ( t θ ) o 4 8 ( 8 ) ( t ) θ ( 4 ( ) 4) Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge 8 o 7
9 θ ( 46.9) 4 8 ( ) θ (.7) K ( ) (.7595) 4 t t 4 4 θ ( t ) θ ( ) 4 8 ( ) ( t ) θ 655.6K ( (.8869) 4) ( ) 4 8 ( ).67.6 θ θ (.8869) (.6) 4 Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge 9 o 7
10 θ (.958) K ( 48) θ K Te results rom Heun s metod re compred wt ect results n Fgure. Te ect soluton o te ordnr derentl equton s gven b te soluton o nonlner equton s θ.959ln.859 tn θ Te soluton to ts nonlner equton t t 48 s s θ ( 48) K (.θ).67 t. 98 Temperture θ (K) Ect Tme t(sec) Fgure Heun s metod results or derent step szes. Usng smller step sze would ncrese te ccurc o te result s gven n Tble nd Fgure below. Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge o 7
11 Tble Eect o step sze or Heun s metod Step sze θ ( 48) E t t % Temperture θ (48) Step sze -4 Fgure Eect o step sze n Heun s metod. In Tble Euler s metod nd te Runge-Kutt nd order metod results re sown s uncton o step sze Tble Comprson o Euler nd te Runge-Kutt metods Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge o 7
12 Step sze θ (48) Euler Heun Mdpont Rlston wle n Fgure 4 te comprson s sown over te rnge o tme. θ(k) Temperture Anltcl Mdpont Rlston Heun 6 Euler Tme t (sec) Fgure 4 Comprson o Euler nd Runge Kutt metods wt ect results over tme. Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge o 7
13 How do tese tree metods compre wt results obtned we ound ( ) drectl? O course we now tt snce we re ncludng te rst tree terms n te seres te soluton s polnoml o order two or less (tt s qudrtc lner or constnt) n o te tree metods re ect. But or n oter cse te results wll be derent. Let us te te emple o d d e I we drectl nd ( ) were 5. te rst tree terms o te Tlor seres gves! ( ) ( ) ( ) e ( ) 5e 9 For step sze o. usng Heun s metod we nd (.6). 9 Te ect soluton gves e 4e (.6) (.6) (.6) e 4e.969 Ten te bsolute reltve true error s t % Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge o 7
14 For te sme problem te results rom Euler s metod nd te tree Runge-Kutt metods re gven n Tble. Tble Comprson o Euler s nd Runge-Kutt nd order metods (.6) Ect Euler Drect nd Heun Mdpont Rlston Vlue t % Append A How do we get te nd order Runge-Kutt metod equtons? We wrote te nd order Runge-Kutt equtons wtout proo to solve (A.) d d ( ) s (A.) were (A.) (A.b) ( ) ( p q ) nd Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge 4 o 7
15 p (A.4) q Te dvntge o usng nd order Runge-Kutt metod equtons s bsed on not smbolcll n te ordnr derentl equton vng to nd te dervtve o So ow do we get te bove tree Equtons (A.4)? Ts s te queston tt s nswered n ts Append. Wrtng out te rst tree terms o Tlor seres re (A.5) d d! d d O were Snce d d ( ) we cn rewrte te Tlor seres s Now ( ) ( ) O( ) (A.6)! Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge 5 o 7
16 Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge 6 o 7 d d. (A.7) Hence! O d d O (A.8) Now te term used n te Runge-Kutt nd order metod or cn be wrtten s Tlor seres o two vrbles wt te rst tree terms s q p O q p (A.9) Hence O q p O q p (A.) Equtng te terms n Equton (A.8) nd Equton (A.) we get p
17 q Source URL: ttp://numerclmetods.eng.us.edu/ Slor URL: ttp:// Attrbuted to: Unverst o Sout Flord: Holstc Numercl Metods Insttute Slor.org Pge 7 o 7
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