Ordinary Differential Equations. Orientation. Lesson Objectives. Ch. 25. ODE s. Runge-Kutta Methods. Motivation Mathematical Background
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1 Ordar Deretal Equats C. 5 Oretat ODE s Mtvat Matematcal Bacgrud Ruge-Kutta Metds Euler s Metd Hue ad Mdpt metds Less Objectves Be able t class ODE s ad dstgus ODE s rm PDE s. Be able t reduce t rder ODE s t a sstem rst rder ODE s. Uderstad te vsual represetats Euler s metd. Kw te relatsp Euler s Metd t te Talr seres epas ad te sgt t prvdes regardg te errr te metd Uderstad te derece betwee lcal ad glbal trucat errrs r Euler s metd.
2 Ordar Deretal Equats: Mtvat Ver Cmm Egeerg Fudametal laws are based cages pscal prpertes Q - dt/d Furer s Law F d/dt (mv Newt s d law Ma ODEs ca be slved aaltcall wever mre cmple es must be attaced umercall Deretal Equats: Classcat Order a deretal Equat. Ordar vs. Partal deretal equats. Lear/N-lear ' m c F(t T T u u u v F ( t ODEs Numercal Sluts Ccetrate st rder ODE s because ger rder ODE s ca be reduced t a set st rder ODEs st Order ODE F( ' d Order ODE F( ' ' ' e cs(
3 Ordar Deretal Equats Reducg ger rder deretal equats t a sstem rst rder equats: m c Dee a ew varable d dt Substtute t te rgal DE m c Ordar Deretal Equats Reducg ger rder deretal equats t a sstem rst rder equats: m c d dt d c dt m I geeral a t rder ODE ca be reduced t st rder ODEs (wt apprprate budar r tal cdts ODEs Numercal Sluts Ital Value Prblems: all cdts are speced at te same value te depedet varable (t r. Prvde a uque slut (r a t rder deretal equat cdts are requred. Budar Value Prblems: cdts are speced at deret values te depedet varable I.e. ( & (43 3
4 Aswer te llwg Wat s (are te depedet varable(s? Wat s (are te depedet varable (s? Is ts a ODE r PDE? Wat rder s ts deretal equat? Is ts lear r lear? dc d C dt d Leard Euler ( Curtes Wpeda Eclpeda Ruge-Kutta Metds CH 5 Slve ODEs te rm: d ( d Ca be slved Numercall usg: φ φ slpe estmate step sze curret value te depedat varable estmate depedat varable ver dst. H Frmula ca be appled step b step t trace ut te slut trajectr. φ errr 4
5 Euler s Metd Te rst dervatve prvdes te slpe at d ' φ ( d Hece ( Euler s Metd Nte: te slpe at te begg te terval s tae as te average slpe ver te etre terval Euler s Metd eample Use Euler s metd t umercall tegrate rm t 4 wt a step sze.5. Te tal cdt at s. Euler s Metd Errr Assessmet Surces Errr:. Trucat Talr Seres. Rud-O sgcat Dgts Trucat Errr: parts. Lcal metd applcat ver step. Glbal accumulated addtve errr ver multple applcats Errr Ttal Errr Trucat Errr RO Errr Number steps 5
6 Euler s Metd Errr Assessmet Lcal Trucat Errr: Frst derve Euler s metd rm T-S Epas t represet: ' ( wt ( ' '! ''! '... '(! Net term (... Nw let ' ( Euler s Metd E a O ( R (! O ( Lcal trucat Errr Euler s Metd Errr Assesmet Ntes: Ts s l te lcal trucat errr Te glbal trucat errr s O( I te uct s a rst rder plmal te metd s eact st Order Metd Te errr patter lds r ger rder metds ( t rder metd Tat s: Te eld eact results r t rder plmal Lcal trucat errr s O( Glbal trucat errr s O( Matlab Pseudcde r Euler s Metd set tegrat rage talze varables set step sze lp t geerate arra lp t mplemet Euler s Metd dspla results 6
7 We ave leared Hw t class deretal equats Hw t reduce t rder ODE s t a sstem st rder ODE s. Te vsual represetat Euler s metd. Te relatsp betwee te Talr seres epas ad Euler s Metd Te derece betwee glbal ad lcal trucat errr Euler s Metd. Euler s Metd Bed Errr Cvergece: I te absece Rud- Errrs ur umercal slut appraces te eact slut as te step sze s reduced t s sad t be cverget Stablt: Depeds te metd ad te deretal equat 7
8 8 Euler s Metd Stablt A umercal metd s ustable te errr grws wtut bud (e.g. epetal grwt r a prblem wc te eact slut s buded. Ca deped te metd as well as te deretal equat. Eample: d d e ( ( ( ( ( ( ( Euler s metd s cdtall stable r: Euler Metd Euler s Metd Stablt < Ts Imples A umercal Metd s ucdtall stable t s stable r a values ad ter parameter s te deretal equat Hue s Metd Predctr Crrectr Apprac ( ' ( Predctr Equat. Beg as wt Euler:. Use t estmate slpe at te ed te terval 3. Calculate a average slpe ( ( ' 4. Etraplate learl rm t ( ( Crrectr Equat
9 Hue s Metd Predctr Crrectr Apprac Slpe ( Slpe ( Predctr Step Average Slpe ( ( Crrectr Step Use average slpe t Obta ew estmate Hue s Metd Iterat step Sce s bt sdes te crrectr equat t ca be appled teratvel as: ( ( ' ( ( Nte: Ts teratve prcedure des t cverge te true aswer Cverges t a te trucat errr Hue s Metd Eample Slve: ' Subject t te I.C. ( at.4 wt.4 usg Heu s metd. Beg wt Predctr Equat ( r tal cdts: ( ( (. Calculate a average slpe '.5 ( ( '.5 ( ( ( (( (.4. '.5 4. Use crrectr equat.5( ( ( (
10 Hue s Metd Eample Slve: ' Subject t te I.C. ( at.4 wt.4 usg Heu s metd Eact Slut: e e.4 At.4 e.4 e. 73 True Errr E t.73.8 % 3%.73 Metd s eact r d rder plmals d rder accurate Lcal trucat errr s O( 3 Glbal trucat errr s O( Hue s Metd Eample Slve: ' Subject t te I.C. ( at.4 wt.4 usg Heu s metd Eact Slut: e e.4 At.4 e.4 e. 73 True Errr E t.73.8 % 3%.73 Metd s eact r d rder plmals d rder accurate Lcal trucat errr s O( 3 Glbal trucat errr s O( Cmpare t Euler? Hue s Metd Matlab Cde Eample See matlab cde Nte tat s l a uct te depedet varable tere s eed t terate ad te llwg equat lds r Hue s metd: ( ( Drectl related t te trapezdal rule
11 φ Ruge-Kutta Metds CH 5 Ca aceve Talr Seres accurac wtut evaluatg ger rder dervatves. Geeral rm: ( ( φ - Icremet uct & s le a slpe ver te terval φ a a... a ( a s are cstats & s are recurrece relatsps Euler s metd φ Ruge-Kutta Metds CH 5 Ca aceve Talr Seres accurac wtut evaluatg ger rder dervatves. Geeral rm: ( ( φ - Icremet uct & s le a slpe ver te terval φ a a... a ( ( p q p q a s are cstats & s are recurrece relatsps Euler s metd 3 ( q ( p q q... q ( Ruge-Kutta Metds T Determe te al rm (. Select. Evaluate a sp sq s b settg te geeral rm equal t terms te T-S epas. 3. Fr lw-rder rms Number terms rder te metd Lcal trucat errr s O( Glbal trucat errr s O(
12 d - Order Ruge-Kutta Metds Geeral Frm: ( a a ( ( p q B settg ( equal t a T-S epas trug te d rder term we ca slve r a a p q a a a q a p / / ( a a 3 Eqs & 4 uws p ( Spec a value / a q / ( a *Sce tere are a te umber cces r a tere wll be a te umber d rder R-K Metds See B 5. Tet d - Order Ruge-Kutta Metds A Hue Metd wtut terat (a ½ : a ½ p q ( slpe at start terval slpe at ed terval ( Glbal Trucat Errr ~ O( a a q p / / ( a ( a d - Order Ruge-Kutta Metds B Mdpt Metd (a : a p / q / ( (.5.5 Glbal Trucat Errr ~ O( a a q p / / ( a ( a
13 d - Order Ruge-Kutta Metds C Ralst s Metd (a /3 : a /3 p 3/4 q 3/4 3 3 ( ( Glbal Trucat Errr ~ O( a a q p / / ( a ( a 4 t - Order Ruge-Kutta Metds Classcal 4 t rder RK Metd mst cmml used RK metd ( φ Slpe Estmates: ( ( ( ( 3 Glbal Trucat Errr ~ O( φ 4 / 4 t - Order Ruge-Kutta Metds Eample: Use classcal RK4 t r - ad.4 e Recall te eact slut s: (.4.73 RK4 Slut: Ital Cdts ( (.5.5 (.4 / (. 3 (.5.5 (.4 / ( (.5(.( (.4 (.6( ( ( ( (. (
14 4 t - Order Ruge-Kutta Metds Eample: Use classcal RK4 t r - ad.4 Errr: E t %.%.73 See Matlab Sample Matlab RK4 metd Metd Cmpars Hger rder metds prduce better accurac Ert r te ger rder metds s smlar t lw-rder metds (muc te ert ges t evaluatg te uct Classcal 4 t rder RK s mst wdel used as t prduces accurate results wt reasable ert. Sstems Equats Recall A t rder ODE ca be represeted as a sstem st rder ODEs d d d d M d d (... (... (... T slve te sstem requres tal cdts at 4
15 5 Sstems Equats RK4 Eample ( ( d d d d Fr eample: ( ( z z d dz z d d 4cs 3 Subject t tal cdts ( ( Y Y Sstems Equats RK4 Eample Slve r slpes j t value r te jt depedat varable Fr RK-4 3 ad 4 wle j umber depedat varables Sstems Equats RK4 Eample ( ( ( ( Slve r slpes Start wt Te tal cdt
16 Sstems Equats RK4 Eample ( ( Sw Matlab Sstems Equats RK4 Eample d d d 7 d ( 4 ( Matlab ODE slvers ODE3 ad ODE45 are RK slvers tat cmbe d ad 3 rd rder RK ad 4 t ad 5 t rder RK metds. See Capter 8 Palm Tet. 6
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