Runge-Kutta 4th order (RK4) — derivation: Foundations

4.8.24 · D1 · Maths › Numerical Methods › Runge-Kutta 4th order (RK4) — derivation

Yeh page un saari vocabulary pieces ko build karta hai jinpar parent note RK4 derivation rely karta hai — shuru karte hain un cheezein se jo ek 12-saal ka bachcha pehle se jaanta hai, aur exactly ek nayi idea ek baar mein add karte hain. Yahaan kuch bhi assumed nahi hai; agar koi symbol dikhta hai, toh usse ek line pehle define kar diya gaya hai.

1. Ek function $f$ aur notation $f (x, y)$

Letter $x$ horizontal position hai (kitna right chale hain hum). Letter $y$ vertical position hai (height). Toh $f (x, y)$ ek rule hai jo, jahan tum ek map par ho (ek right-distance aur ek height), ek akela number return karta hai.

Hamare topic mein woh returned number ek slope hai — us jagah par path kitna tilted hai. Figure 1 dekho: plane ke har point par, $f$ ek chota arrow paint karta hai jo dikhata hai kidhar jhukna hai.

Figure — Runge-Kutta 4th order (RK4) — derivation

2. Slope, aur symbol $y^{'}$

Ek ramp imagine karo. Agar har $1$ metre chalte right par $2$ metre upar jaate ho, toh slope $2$ hai. Hum is "rate of rising" ko ek shorthand symbol dete hain:

Toh hamare topic ki saari problem ek sentence hai: $y^{'} = f (x, y)$ jo padha jaata hai: "meri curve ka slope, kisi bhi point par, machine $f$ deta hai." Hum slope-rule har jagah jaante hain; hum curve khud nahi jaante. Hamaara kaam curve ko reconstruct karna hai.

3. The Initial Value Problem (IVP) aur $y (x_{0}) = y_{0}$

Slope ko har jagah jaanna abhi bhi kaafi nahi hai — infinitely many curves ka same slope-field hota hai, bas upar ya neeche shifted. Humein ek starting point pakka karna hoga.

Ek slope-rule plus ek starting point ko Initial Value Problem (IVP) kehte hain. Figure 1 ke arrows slope-rule hain; ek single dot curve ko pin karta hai. Us dot se, arrows batate hain aage kahan jaana hai.

4. Subscripts $n$ , $n + 1$ — steps mein aage badhna

Hum curve ko ek smooth swoop mein nahi draw kar sakte; hum ise ek chote hop mein lete hain.

Toh $y_{0} \to y_{1} \to y_{2} \to \dots$ fog ke paas stepping-stones ki ek chain hai.

5. Step size $h$

Figure 3 same step dikhata hai chote vs bade $h$ ke saath: chota step true curve ko hug karta hai, bada step drift kar sakta hai.

6. Integral $\int_{x_{n}}^{x_{n} + h} f d x$

Parent note exact answer ko ek integral ki tarah likhta hai. Yahaan woh symbol zero se hai.

Toh exact step hai "nayi height = purani height + total rise": $y (x_{n} + h) = y_{n} + \int_{x_{n}}^{x_{n} + h} f (x, y (x)) d x .$

7. Big-O notation: $O (h^{5})$ , $O (h^{4})$

Local error $O (h^{5})$ : ek single step mein ki gayi galti.
Global error $O (h^{4})$ : poore interval ko cross karne ke baad kul galti. Cross karne mein lagbhag $1/ h$ steps lagte hain, toh per-step $O (h^{5})$ errors $O (h^{4})$ mein pile up ho jaate hain — ek power kho jaata hai. (Aur details Local vs Global Truncation Error mein.)

8. Taylor series — "true answer" jo hum chase karte hain

Parent note RK4 ko real curve ki Taylor series se match karne ke liye tune karta hai. Yeh kya hoti hai?

Yahaan $y^{''}$ ("y double-prime") slope ka slope hai — tilt khud kitni tezi se change ho rahi hai (curve ka bending). RK4 ki cleverness: yeh un terms ko bina kabhi $y^{''}, y^{'''}, \dots$ compute kiye reproduce karta hai — yeh unhe $f$ ke extra evaluations se fake karta hai. $y^{''}, y^{'''}, \dots$ directly compute karne wala version dekhne ke liye Taylor Series Methods dekhein.

9. Stage slopes $k_{1}, k_{2}, k_{3}, k_{4}$

Tumhe inhe yahaan derive karne ki zaroorat nahi — parent note karta hai. Tumhe sirf $k_{i}$ ko padna hai jaise "ek slope-peek se ek chosen sample point par step ka rise guess." Modified Euler / Heun's Method sirf do peeks use karta hai (edges); RK4 chaar use karta hai.

Prerequisite map

Equipment checklist

Read the prompt, answer in your head, then reveal.

Is topic mein

f (x, y)

kya output karta hai?

Ek slope — curve

(x, y)

point par kitni tilted hai.

y^{'}

ka plain words mein kya matlab hai?

Curve

y

ka slope: height kitni tezi se change hoti hai per step right.

f

ko do inputs

x

aur

y

kyun chahiye?

Slope dono par depend kar sakta hai — tum horizontally kahan ho aur tum pehle se kitne upar ho.

Ek slope-field ko ek single curve mein kya convert karta hai?

Initial condition

y (x_{0}) = y_{0}

— ek pinned starting dot.

Subscripts

n

aur

n + 1

kya track karte hain?

Hum kaunse hop par hain;

n + 1

agla stepping-stone hai.

Step size

h

kya hai?

Har step mein hum kitna right hop karte hain:

x_{n + 1} = x_{n} + h

Har

k_{i}

h

se multiply kyun kiya jaata hai?

Slope rise hai per unit run;

h

ke run par rise

slope \times h

hoti hai.

Integral

\int f d x

geometrically kya represent karta hai?

Slope curve ke neeche ka area = step par kul height gained.

O (h^{4})

tumhe kya batata hai?

h

ko half karne par error

2^{4} = 16

se shrink hota hai; bada power = zyada accurate.

Local aur global error mein kya fark hai?

Local = ek step ki galti (

O (h^{5})

); global =

\sim 1/ h

steps ke baad accumulated (

O (h^{4})

Yahaan Taylor series kya hai?

y^{'}, y^{''}, y^{'''}, \dots

se rebuild ki gayi exact curve — woh target jise RK4

h^{4}

tak match karta hai.

Har

k_{i}

kya hai?

Ek chosen sample point par ek slope-peek se step ke rise ka estimate.

Connections

Euler's Method — sirf $k_{1}$ use karta hai (ek left-edge peek).
Modified Euler / Heun's Method — do peeks (dono edges).
Simpson's Rule — $1, 2, 2, 1$ blending weights ka source.
Taylor Series Methods — $y^{''}, y^{'''}$ directly compute karta hai; RK4 unhe fake karta hai.
Local vs Global Truncation Error — $O (h^{5})$ vs $O (h^{4})$ ki kahani.
Adaptive Step Size (RKF45) — $h$ automatically choose karta hai.