Adaptive step-size — RK45, error control — Hinglish notes

4.8.25 · Maths › Numerical Methods

WHAT we are trying to do

WHY do orders? Ek akela method aapko ek number $y_{n + 1}$ deta hai, lekin yeh jaanne ka koi tarika nahi ki woh kitna accha hai. Agar aapke paas ek 4th-order estimate $y^{(4)}$ aur ek 5th-order estimate $y^{(5)}$ same stages $k_{i}$ se hain, toh $ε = y^{(5)} - y^{(4)}$ essentially 4th-order result ka local error hai — almost koi extra cost nahi. Isko embedded pair kehte hain.

HOW the stages work (the "embedded" idea)

Ek Runge–Kutta method $s$ slope samples se $y_{n + 1}$ banata hai:

y_{n+1} = y_n + h\sum_{i=1}^{s} b_i k_i$$ Ek **embedded pair** same $k_i$ share karta hai lekin do weight rows $b_i$ (order 5) aur $b_i^{*}$ (order 4) use karta hai: $$y^{(5)} = y_n + h\sum_i b_i k_i,\qquad y^{(4)} = y_n + h\sum_i b_i^{*} k_i$$ $$\boxed{\;\varepsilon = y^{(5)} - y^{(4)} = h\sum_i (b_i - b_i^{*})\,k_i\;}$$ > [!intuition] Yeh "free" kyun hai > Kisi bhi RK method ka expensive part function evaluations $k_i$ hain. Dono answers inhe **sab** reuse karte hain, isliye error estimate sirf ek sasta dot product cost karta hai, extra $f$ calls nahi. --- ## DERIVING the step-update formula from scratch Hum assume karte hain ki lower-order method ki error leading term ki tarah scale hoti hai: $$\varepsilon(h) \approx C\,h^{p+1}, \qquad p = 4 \text{ for RK45.}$$ **WHY:** order-$p$ method ki local truncation error apne pehle uncancelled Taylor term se dominate hoti hai, jo $\propto h^{p+1}$ hai. Hum ek naya step $h_{\text{new}}$ chahte hain jiska error tolerance ke barabar ho: $$C\,h_{\text{new}}^{\,p+1} = \text{tol}.$$ Unknown constant $C$ ko eliminate karne ke liye dono equations ko divide karo: $$\frac{\text{tol}}{\varepsilon} = \frac{C h_{\text{new}}^{p+1}}{C h^{p+1}} = \left(\frac{h_{\text{new}}}{h}\right)^{p+1}.$$ $h_{\text{new}}$ ke liye solve karo: $$\boxed{\,h_{\text{new}} = h \left(\frac{\text{tol}}{\varepsilon}\right)^{1/(p+1)}\,}$$ > [!formula] Practical step controller > $$h_{\text{new}} = h \cdot \min\!\Big(f_{\max},\;\max\big(f_{\min},\; S\,(\text{tol}/\varepsilon)^{1/(p+1)}\big)\Big)$$ > - $S \approx 0.8$–$0.9$ : **safety factor** (tol hit karne par gamble mat karo). > - $f_{\min}\approx 0.2,\; f_{\max}\approx 5$ : clamp taaki $h$ kabhi wildly jump na kare. > - $p+1 = 5$ for RK45. > **Safety factor kyun?** $\varepsilon$ khud ek estimate hai; thoda *tol ke neeche* aim karo taaki retried step pehli baar hi accept ho jaye. ![[4.8.25-Adaptive-step-size-—-RK45,-error-control.png]] --- ## Worked Example 1 — accept and grow > [!example] Ek adaptive step > Suppose ek step par aapne $h = 0.10$ use kiya, $\varepsilon = 1\times10^{-7}$ mila, tol $= 1\times10^{-6}$, $S=0.9$, $p=4$. > > **Step 1 — accept?** $\varepsilon < \text{tol}$ ✔ → is step ko accept karo. > *Yeh step kyun?* Error budget ke andar hai, toh answer kaafi accha hai. > > **Step 2 — new $h$:** ratio $= \text{tol}/\varepsilon = 10$. > $$h_{\text{new}} = 0.9 \times 0.10 \times 10^{1/5} = 0.09 \times 1.585 = 0.143.$$ > *Yeh step kyun?* Hamare paas spare error thi, isliye hum $h$ **grow** karte hain tezi se kaam karne ke liye — lekin sirf $10^{1/5}$ se, 10 se nahi, kyunki error $h^5$ ki tarah scale hoti hai. ## Worked Example 2 — reject and shrink > [!example] Ek rejected step > Ab $h = 0.10$, $\varepsilon = 4\times10^{-6}$, tol $= 1\times10^{-6}$. > > **Step 1 — accept?** $\varepsilon > \text{tol}$ ✗ → **reject** karo, step redo karo. > *Yeh step kyun?* Step bahut inaccurate hai; hum $t$ ko aage nahi badha sakte. > > **Step 2 — new $h$:** ratio $= 0.25$. > $$h_{\text{new}} = 0.9\times0.10\times0.25^{0.2} = 0.09\times0.758 = 0.068.$$ > *Yeh step kyun?* $h$ shrink karo taaki **retried** step ki predicted error $\approx C h_{\text{new}}^5 = $ tol ho. Hum same $t_n$ se $0.068$ ke saath retry karte hain. ## Worked Example 3 — clamp in action > [!example] Forecast-then-verify > $h = 0.05$, $\varepsilon = 1\times10^{-12}$, tol $=1\times10^{-6}$, $f_{\max}=5$. > Raw factor $= 0.9\,(10^{6})^{1/5} = 0.9\times 15.8 = 14.2$. > **Forecast:** yeh $h$ ko 14× enlarge kar deta — dangerous! > **Verify:** controller $f_{\max}=5$ par clamp karta hai, toh $h_{\text{new}} = 0.25$. > *Yeh step kyun?* $h^5$ model sirf locally trustworthy hai; ek 14× jump aise region mein land kar sakta hai jo model ne kabhi nahi dekha. Clamping humein safe rakhta hai. --- > [!mistake] Steel-manned errors > **Mistake A:** "Error $h^4$ ki tarah scale hoti hai toh exponent $1/4$ use karo." > *Yeh sahi kyun lagta hai:* RK45 ka *result* order 4 (ya 5) ka hai. **Fix:** **local** error per step $\sim h^{p+1}=h^5$ hai, isliye exponent $1/(p+1)=1/5$ hai. $1/4$ use karne se controller over-react karta hai. > > **Mistake B:** "Reject karne ke baad, chhote $h$ ke saath $t$ aage badha do." > *Yeh sahi kyun lagta hai:* Aapne "kaam kiya," toh ise throw away karna wasteful lagta hai. **Fix:** ek rejected step matlab $y_{n+1}$ untrustworthy hai — aapko **$t_n$ par hi rehna hai** aur naye $h$ ke saath recompute karna hai. > > **Mistake C:** Sirf absolute tolerance use karna. > *Yeh sahi kyun lagta hai:* simpler hai. **Fix:** agar $|y|$ bada ho jaaye, ek fixed atol impossibly strict ho jaata hai; $\text{rtol}\,|y|$ mix karo taaki tolerance solution ke saath scale kare. > > **Mistake D:** "Error estimate $y^{(5)}$ ki error hai." > *Yeh sahi kyun lagta hai:* aap $y^{(5)}$ ke saath aage badhte ho (local extrapolation). **Fix:** $\varepsilon$ *4th-order* error estimate karta hai; yeh dono mein se worse ko bound karta hai, jo conservative hai — woh theek hai. --- > [!recall]- Feynman: ek 12-saal ke bachche ko explain karo > Sochiye aap ek car drive kar rahe ho aur aap winding road par bina crash ke rehna chahte ho. Lambi seedhi jagah par aap speed up karte ho; sharp turns ke paas slow down karte ho. Yeh jaanne ke liye ki turn kitna sharp hai, aap **do cameras** se dekhte ho — ek rough, ek sharp — aur dekhte ho ki unki pictures kitni alag hain. Agar woh barely alag hain, road seedhi hai: floor it (bada step). Agar woh bahut disagree karte hain, turn aa raha hai: brake lagao (chhota step). RK45 bilkul yahi hai: do answers, unhe compare karo, aur apni speed (step size) choose karo taaki aap kabhi badi galti na karo. > [!mnemonic] Controller yaad rakho > **"Tol Over Error, to the Fifth-root"** → $h_{\text{new}} = h\,(\text{tol}/\varepsilon)^{1/5}$. > Aur **"Reject → Repeat, Accept → Advance."** --- ## #flashcards/maths RK45 har step par do solutions kyun compute karta hai? ::: *Same* stage evaluations $k_i$ se ek free local-error estimate $\varepsilon = y^{(5)}-y^{(4)}$ paane ke liye. "Embedded" Runge–Kutta pair kya hota hai? ::: Do methods alag order ke jo same $k_i$ share karte hain lekin alag weight rows $b_i,\,b_i^*$ rakhte hain. Order-$p$ method ki local error kaise scale hoti hai? ::: $C\,h^{p+1}$. Naya step size derive karo. ::: $\varepsilon\approx Ch^{p+1}$ aur $\text{tol}=Ch_{\text{new}}^{p+1}$ se, $C$ cancel karne ke liye divide karo: $h_{\text{new}}=h(\text{tol}/\varepsilon)^{1/(p+1)}$. RK45 (order 4 controlled) ke liye $h$ update karne ka exponent kya hai? ::: $1/(p+1)=1/5$. Rejected step ke baad kya karte hain? ::: $t_n$ par ruko, chhote $h_{\text{new}}$ ke saath recompute karo; $t$ aage mat badha. Safety factor $S\approx0.9$ ka purpose kya hai? ::: Thoda tol ke neeche aim karo taaki retried/next step usually pehli baar pass ho jaye. $h_{\text{new}}$ ko $f_{\min},f_{\max}$ se kyun clamp karte hain? ::: $h^{p+1}$ model sirf locally valid hai; clamps wild jumps rokti hain. Standard tolerance formula kya hai? ::: $\text{tol}=\text{atol}+\text{rtol}\,|y_n|$. Error estimate essentially "free" kyun hai? ::: Yeh saare expensive $f$-evaluations reuse karta hai; sirf ek sasta weighted sum add hota hai. --- ## Connections - [[Runge–Kutta Methods (RK4)]] — RK45 ka fixed-step parent. - [[Local vs Global Truncation Error]] — kyun $h^{p+1}$ step controller govern karta hai. - [[Order of a Numerical Method]] — exponent mein $p$ define karta hai. - [[Stiff ODEs and Stability]] — jab adaptive explicit methods phir bhi struggle karte hain. - [[Taylor Series Expansion]] — leading error term ki origin. - [[Dormand–Prince (RK45 used by scipy solve_ivp)]] — standard coefficient table. ## 🖼️ Concept Map ```mermaid flowchart TD FIXED[Fixed step h wasteful] -->|motivates| ADAPT[Adaptive step-size] ADAPT -->|needs| ESTERR[Free error estimate] RK45[RK45 embedded pair] -->|provides| ESTERR RK45 -->|shares| STAGES[Same slopes k_i] STAGES -->|weighted by| Y4[Order-4 estimate] STAGES -->|weighted by| Y5[Order-5 estimate] Y5 -->|difference| EPS[Error epsilon] Y4 -->|difference| EPS TOL[tol = atol + rtol times y] -->|compare with| DECIDE{epsilon <= tol} EPS -->|compare with| DECIDE DECIDE -->|yes| ACCEPT[Accept step] DECIDE -->|no| REJECT[Reject and shrink h] EPS -->|assume C h^p+1| UPDATE[h_new formula] TOL -->|eliminate C| UPDATE UPDATE -->|with safety S| CTRL[Step controller] ```