Runge Kutta methods of order 2

fEmRecall EDTaylor methods of order 28

error 0 he

Ideame Find a di B s t

T ti Wi a Htt di ytBand error stays Ochi

The derivation requires 2 Ingredientsme

Engineered Taylor polynomialsfor functions of 2 Variables s

f Et d g y 1 Bi

i

Effy 5

error term Ii OEFwhere 3M between ft g R t 14 yep

ingredient Ohari Rule aa

Ct g effect yes H

Felt g Ightg offbut THE g Ht g h f CE g

Ft2

so TGKGyj f

hzffzy.frzzRewriting multiplying it by a e

aaq.stmaat.se

a

agEEgEt7Q

Since we want THE a a Atta ytp

we just match coefficients offt Idg in And we

solve for a di Pi 8

gL hz.fHy a

So now

ffthzsythzfft.gs THEyR E gtEHtyD

eyehkfltoBEE.ro fhqhqtyD3IgEy

0 h7ih all partialsare bounded

So we replace Ta CE g byf Ethel y theft yD

to getsNo L

ftp.IFENG

k.witEatiwToi

Ringe Kutta methodof order 2also known agoomidp.intmethod

Anotheroch.TWhodisgivenbg

w hzq fftin wuhHtiioilw

b y atfti.wu.MEmodified at withEuler Method 1st approx

of win

x Averageof derivativesHopes

Idea behind this modification 3mm

Wi t h Nti Wi writ is the

estimate Crane Euler's methodof YETIRather than use Tix directlywe plug it back into

Htin Toit estimate ofslopeat titiand average it with f ti Wi

to getwit Wit LEHI wi tf tit ifD

2where wT With Ati Wi

Example of Higher order RKmethod e

RK order 4 8rn

Wo L est Tig

a iE9 wifi

kg LG ti tha wiTgkby h f t s Wi t B

WE11 Wi t f 2kt 2kg kg

Err h

g

swmmayoyournumerial.memethodsforIVPS e

flt g te La b

yeas a

0 h

w wi hfCtisw

Higher order Taylormethods 06h

If Eui th Ta Cti WiYee Tk Cti Wi Nti Wi 1 hzfttiswD

ffkti.wilt i.tk dnYti

TI Rurge kuttam

ethod.sca0hDNo L

Wait Wit ht titty withAtaD

II b Order 4 Othsee before

Other modifications

e.g modified Euler

No LWiti Wi t hz f ti Wi tHtifwwT withhGis