3. 3E: Runge-Kutta Usuli (Mashqlar) Matematika



Download 72,08 Kb.
bet15/15
Sana19.03.2022
Hajmi72,08 Kb.
#501496
1   ...   7   8   9   10   11   12   13   14   15
Qaror
The step size (h) goes down by (frac<1><3>) so the error is ( > ight)^<4>=frac<1><81>) of the original error.Exercise 19
You’ve estimated the error of a Forward Euler method on an interval ([0,10]) with (20) time steps. If you increase to (100) time steps, by what factor will the error go down?
Qaror
The step size (h) goes down by (frac<1><5>) so the error is ( left<(frac<1><5>> ight)^<2>=frac<1><25>) of the original error.
Exercise 20
Suppose (x(t)) solves the differential equation (x'=-2tx^2) with initial condition x(1) = 1. Solve the above IVP for x(2) using:
a) Euler method with step size (0.5)
b) Analytically.
Qaror
a) We have (f(t,x)=-2tx^2) . With (x(1)=1) , we have (f(1,1)=-2) so we adjust (x) by ((0.5)cdot(-2)=-1) . Thus we estimate (x(1.5)=0) . Repeat the process, with (f(1.5,0)=0) causing no adjustment. Our estimate is (x(2)=0) .
b) This is a separable equation. It separates to [int frac=int -2tdt ,.] The implicit form of the solution is [frac<-1>=-t^2+c ,.] The explicit form is [x(t)=frac<1> ,.]
We plus in our initial condition, (frac<1><1+c>=1) so (c=0) and our solution is [x(t)=frac<1> ,.] Therefore (x(2)=frac<1><4>) .
Exercise 21
Use the explicit Euler method to plot the estimate (y(2)) if (y(t)) is the solution to (y'=t-y^3) with (y(0)=1) . Use (10) steps, (20) steps, and (30) steps and plot all three.
Qaror

Use the sample code provided in the chapter and add a line to plot the result.
Exercise 22
Use the Explicit Euler and Runge-Kutta methods to estimate (y(2)) if (y(t)) is a solution to (y'=frac<1>) with (y(0)=1) and (10) time steps.
Qaror

Combine the sample codes for Runge-Kutta and Euler Explicit from the chapter notes.

2.4. Implicit Runge-Kutta schemes¶
We have discussed that explicit Runge-Kutta schemes become quite complicated as the order of accuracy increases. Implicit Runge-Kutta methods might appear to be even more of a headache, especially at higher-order of accuracy (p) . We will give a very brief introduction into the subject, so that you get an impression.
Generally speaking, RK methods can be defined as follows:
For explicit RK methods one has (h=s-1) . This implies that any (k_i) can be computed explicitly from the knowledge of the previously computed (k_j) with (jAs an example, let us consider some relatively simple implicit RK scheme - we went on Wikipedia and picked one named Qin and Zhang’s two-stage second-order implicit method. It reads:
Let’s implement it for the problem of a body in free fall described by (20). Make sure you understand the implementation of the implicit scheme in the code
Download 72,08 Kb.

Do'stlaringiz bilan baham:
1   ...   7   8   9   10   11   12   13   14   15




Ma'lumotlar bazasi mualliflik huquqi bilan himoyalangan ©hozir.org 2024
ma'muriyatiga murojaat qiling

kiriting | ro'yxatdan o'tish
    Bosh sahifa
юртда тантана
Боғда битган
Бугун юртда
Эшитганлар жилманглар
Эшитмадим деманглар
битган бодомлар
Yangiariq tumani
qitish marakazi
Raqamli texnologiyalar
ilishida muhokamadan
tasdiqqa tavsiya
tavsiya etilgan
iqtisodiyot kafedrasi
steiermarkischen landesregierung
asarlaringizni yuboring
o'zingizning asarlaringizni
Iltimos faqat
faqat o'zingizning
steierm rkischen
landesregierung fachabteilung
rkischen landesregierung
hamshira loyihasi
loyihasi mavsum
faolyatining oqibatlari
asosiy adabiyotlar
fakulteti ahborot
ahborot havfsizligi
havfsizligi kafedrasi
fanidan bo’yicha
fakulteti iqtisodiyot
boshqaruv fakulteti
chiqarishda boshqaruv
ishlab chiqarishda
iqtisodiyot fakultet
multiservis tarmoqlari
fanidan asosiy
Uzbek fanidan
mavzulari potok
asosidagi multiservis
'aliyyil a'ziym
billahil 'aliyyil
illaa billahil
quvvata illaa
falah' deganida
Kompyuter savodxonligi
bo’yicha mustaqil
'alal falah'
Hayya 'alal
'alas soloh
Hayya 'alas
mavsum boyicha


yuklab olish