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