Stiff equations — implicit methods, backward Euler: Visual walkthrough

4.8.26 · D2 · Maths › Numerical Methods › Stiff equations — implicit methods, backward Euler

Pehli line se pehle, teen seedhe words jinpe hum poore waqt tike rahenge:

Step 1 — Test problem ek girta hua curve hai

KYA. Hum ek simple equation padhte hain: $y^{'} = λ y .$

KYUN. Stiff problems pe har method ka behaviour is ek line se decide hota hai, isliye pehle yahi samjhte hain. Zor se padho: "slope equals $λ$ times height." Yahan $λ$ (Greek "lambda") bas ek fixed negative number hai — ek dial. Negative $λ$ matlab: jitna zyada upar ho, utna tezi se giro.

TASVEER. Sahi motion hai $y (t) = y_{0} e^{λ t}$ — ek curve jo height $y_{0}$ se shuru hoti hai aur smoothly $0$ ki taraf khisak jaati hai. $λ$ ko aur negative karo aur girna shuru mein brutally fast ho jaata hai, phir flat ho jaata hai.

Figure — Stiff equations — implicit methods, backward Euler

Step 2 — Hum curve ko continuously follow nahi kar sakte, isliye hop karte hain

KYA. Computer sirf discrete instants $t_{0}, t_{1}, t_{2}, \dots$ pe $y$ store karta hai, jo ek step size $h$ se alag hote hain: $t_{n + 1} = t_{n} + h .$

KYUN. Hum smooth rule $y^{'} = λ y$ ko ek hopping recipe mein badalna chahte hain: abhi ki height di ho, ek hop baad ki height guess karo. Iske liye ek honest tool hai — slope ki definition — rise over run.

TASVEER. Curve pe do dots, $h$ ka horizontal gap. Unke beech ki seedhi line ki slope "rise ÷ run" hai: $slope \approx \frac{y _{n + 1} - y _{n}}{h} .$ Yahan $y_{n}$ ka matlab hai "hop number $n$ pe stored height" — bas ek labelled dot, kuch zyada nahi.

Step 3 — Forward Euler: us slope pe trust karo jahan tum khade ho

KYA. Estimated slope ko purane dot $t_{n}$ pe sahi slope ke barabar set karo: $\frac{y _{n + 1} - y _{n}}{h} = f (t_{n}, y_{n}) = λ y_{n} .$ Nayi height ke liye solve karo: $y_{n + 1} = (1 + hλ) y_{n}$

KYUN. Yeh sabse sasta possible move hai: right side mein sab kuch already known hai, toh $y_{n + 1}$ bina kisi solving ke nikal aata hai. Isse hum explicit kehte hain.

TASVEER. Purane dot pe khado, wahan slope arrow padho, aur ek seedhi line mein pura step $h$ chalo. Kyunki sahi curve us arrow se bend karke door jaati hai, lamba walk overshoot karta hai.

Step 4 — Forward Euler kyun blast karta hai: $1 + hλ$ ki number line

KYA. Stability ka matlab hai $∣1 + hλ ∣ < 1$ . Real $λ < 0$ ke liye $x = hλ$ likho (ek negative number). Chahiye $- 1 < 1 + x < 1$ , yaani $- 2 < x < 0$ . Ab $x < 0$ ko $h$ pe ek bound mein translate karo.

KYUN sign flip hota hai. Hamare paas hai $- 2 < hλ$ . $h$ isolate karne ke liye dono sides ko $λ$ se divide karte hain — lekin $λ$ negative hai, aur inequality ko negative number se divide karne par $<$ palat ke $>$ ho jaata hai. Toh $- 2 < hλ$ ban jaata hai $\frac{- 2}{λ} > h$ . Kyunki $λ < 0$ hai, quantity $\frac{- 2}{λ}$ positive hai aur $\frac{2}{∣ λ ∣}$ ke barabar hai. Isliye: $h < \frac{2}{∣ λ ∣} .$

Yeh kyun matter karta hai. Yeh inequality ek line mein poori stiffness tragedy hai. Agar $λ = - 1000$ hai, tum $h < 0.002$ pe hamesha ke liye chain ho, tab bhi jab curve flat aur boring ho jaata hai.

TASVEER. $1 + hλ$ ko number line pe rakho. Safe zone segment $(- 1, 1)$ hai. $h$ badhne par, point $1 + hλ$ left ki taraf march karta hai, $- 1$ se aage nikal jaata hai, aur wahan land karta hai jahan $∣ g_{FE} ∣ > 1$ ho — ab har hop magnify karta hai aur sign flip karta hai, toh dots infinity tak ping-pong karte hain.

Step 5 — Backward Euler: us slope pe trust karo jahan tum land karte ho

KYA. Wahi slope bridge, lekin sahi slope naye dot $t_{n + 1}$ pe padho: $\frac{y _{n + 1} - y _{n}}{h} = f (t_{n + 1}, y_{n + 1}) = λ y_{n + 1} .$

KYUN. Naya dot wahan hai jahan curve shant hai (flatter). Shant future ki taraf aim karna overshoot rokta hai. Keemnat: $y_{n + 1}$ ab dono sides pe hai — hume iske liye solve karna padega. Yehi implicit ka matlab hai.

TASVEER. Purane arrow ki direction mein nikalne ki jagah hum poochte hain: "kaun si landing height aisi hai ki landing point se wapas khicha hua arrow exactly purane dot ko hit kare?" Hum woh dot choose karte hain jiski apni slope ghar waapas point kare.

Step 6 — Backward Euler hamesha tame kyun rehta hai

KYA. Real $λ < 0$ ke saath, term $- hλ$ positive hai, toh $1 - hλ > 1$ . $1$ se bade number se divide karna $y$ ko shrink karta hai: $\frac{1}{1 - hλ} < 1 har h > 0 ke liye.$

KYUN. Chahe step $h$ kitna bhi bada ho, denominator sirf badhta hai, toh $∣ g_{BE} ∣$ sirf ghatta hai. $h$ pe koi upper limit nahi — stability free hai.

TASVEER. Dono growth factors ko $h$ ke against plot karo ( $λ < 0$ fixed rakh ke). Forward Euler ka $∣1 + hλ ∣$ neeche jaata hai, zero ko touch karta hai, phir safe band se bahar chad jaata hai. Backward Euler ka $∣1/ (1 - hλ) ∣$ $1$ se shuru hota hai aur gently $0$ ki taraf slide karta hai — kabhi band nahi chhodta.

Step 7 — Poori complex tasveer (ek saath har case)

KYA. Real problems mein oscillating parts bhi hote hain, toh $λ$ ek complex number ho sakta hai (ek plane mein ek point: horizontal axis = decay rate $Re λ$ , vertical = wobble rate $Im λ$ ). Sahi solution ke saath honestly compare karne ke liye, exact growth per step ko bhi ek naam do:

$∣ g_{true} ∣$ ko $∣ g_{FE} ∣, ∣ g_{BE} ∣$ se compare karna batata hai ki har method kahaan reality se match karta hai.

TASVEER.

Forward Euler: $∣1 + hλ ∣ < 1$ ek disc hai radius $1$ , centred at $- 1$ . Chhota sa. Left half-plane ka zyaadatar hissa iske bahar hai — yahi stiff $λ$ ke liye instability hai.
Backward Euler: $∣1/ (1 - hλ) ∣ < 1$ woh sab kuch hai jo disc of radius $1$ centred at $+ 1$ ke bahar hai — jo poori left half-plane ko apne andar le leta hai. Yahi A-stable ki definition hai.

Ab plane ko column by column chalte hain taaki koi bhi $λ$ unexplained na rahe.

Definition Plane ka har column, detail mein

$Re λ < 0$ (left half — decaying): sahi solution shrink karta hai ( $∣ g_{true} ∣ = e^{ha} < 1$ ). Backward Euler bhi shrink karta hai ( $∣ g_{BE} ∣ < 1$ ) kisi bhi $h$ ke liye, upar ke box se; forward Euler sirf apni chhoti disc ke andar shrink karta hai.
$Re λ = 0, Im λ \neq = 0$ (imaginary axis — pure oscillation): sahi solution $e^{iω t}$ na badhta hai na ghatta hai, woh spin karta hai $∣ g_{true} ∣ = 1$ ke saath. Backward Euler deta hai $∣ g_{BE} ∣ = \frac{1}{1 - ihω} = \frac{1}{1 + ( hω ) ^{2}} < 1$ — woh oscillation ko damp karta hai instead of preserve karne ke. Stiff safety ke liye achha hai, lekin ek warning: pure oscillators woh amplitude kho dete hain jo unhe rakhni chahiye thi. (Trapezoidal method exactly yahi theek karta hai, axis pe $∣ g ∣ = 1$ deta hai.)
$Re λ > 0$ (right half — genuinely blowing up): sahi solution $e^{λ t}$ bina bound ke badhta hai; ek faithful method ko bhi badhna chahiye. Forward Euler hamesha yahi karta hai: real $λ > 0$ ke liye $∣1 + hλ ∣ > 1$ . Backward Euler thoda aur subtle hai — real $λ > 0$ ke liye, $∣ g_{BE} ∣ = \frac{1}{∣1 - hλ ∣}$ , toh woh tabhi badhta hai jab $0 < hλ < 2$ ; $hλ = 2$ pe denominator ka magnitude $1$ hai aur $∣ g_{BE} ∣ = 1$ , aur $hλ > 2$ ke liye $∣1 - hλ ∣ > 1$ hota hai, jo $∣ g_{BE} ∣ < 1$ banata hai — backward Euler tab galti se ek genuinely exploding solution ko damp karta hai. Yeh A-stability ki honest small print hai: yeh decaying $λ$ ke liye achhe behaviour ki guarantee deta hai, aur woh large-step unstable $λ$ ke liye growth faithfully reproduce na karke kharidta hai.

Definition Degenerate & limiting cases (kabhi skip mat karo)

$λ = 0$ (na decay, na wobble): $g_{FE} = 1$ , $g_{BE} = 1$ , aur $g_{true} = e^{0} = 1$ . Sab boundary pe baithe hain — height constant rehti hai, sahi hai.
$h \to \infty$ (backward Euler, $Re λ < 0$ ): $g_{BE} = \frac{1}{1 - hλ} \to 0$ . Woh sirf bounded nahi rehta, woh snap hokar zero ho jaata hai — yeh stronger property hai L-stability (dekho A-stability and L-stability).
Boundary $∣ g_{BE} ∣ = 1$ : yeh poora circle $∣1 - hλ ∣ = 1$ hai (unit circle centred at $+ 1$ ), koi ek point nahi. Yeh milta hai, for example, real point $hλ = 2$ pe aur, limit $hω \to 0$ mein, imaginary axis pe. Us circle ke bahar har jagah (jo left half-plane ka poora hissa include karta hai) backward Euler strictly tame hai.
Nonlinear $f$ : implicit line rearrange nahi ho sakti; $g (y) = y - y_{n} - h f (t_{n + 1}, y) = 0$ ko Newton's method se solve karo.

Ek-tasveer summary

Do arrows, ek plane. Forward Euler purani slope se chalta hai aur apni chhoti safe disc se bahar jump kar sakta hai. Backward Euler future dot ke liye solve karta hai aur uska safe region poori decaying half-plane ko nigal leta hai.

Recall Feynman: poora walkthrough seedhe shabdon mein

Hamare paas ek dot hai jo floor pe girna chahta hai, aur jitna upar hai utna tezi se girta hai — yahi hai $y^{'} = λ y$ . Computer ise smoothly girte nahi dekh sakta, toh woh width $h$ ke hops leta hai aur agle height ka guess karta hai "rise over run" use karke. Forward Euler padh leta hai ki dot jahan khada hai wahaan kitni tezi se gir raha hai aur ek pura hop seedha chalta hai — lekin sahi girna curve ho jaata hai, toh lamba hop overshoot karta hai, aur fast giraawat ke liye ( $λ$ bahut negative) itna overshoot hota hai ki sign flip ho jaata hai aur blast hota hai jab tak hop microscopic na ho. Backward Euler clever cousin hai: woh poochta hai "mujhe kahan land karna chahiye taaki landing spot se wapas khicha hua fall-speed exactly wahan point kare jahan se main shuru hua tha?" Kyunki woh shant future ki taraf aim karta hai, woh kabhi overshoot nahi karta — multiplier $1/ (1 - hλ)$ hamesha ek shrink hai, kisi bhi hop size ke liye jab $λ$ decay kare. Reason pure geometry hai: number $1 - hλ$ hamesha $1$ ki daayein land karta hai jab $λ$ decay kare, toh uska size $1$ se zyaada hota hai aur usse divide karna shrink karta hai. Agar sahi motion ek pure circle-spin hai (imaginary $λ$ ), backward Euler phir bhi use thoda shrink karta hai — woh wobbles ko damp karta hai jo use "rakhne chahiye" the, yeh safety ki chhoti si keemnat hai. Aur agar sahi motion genuinely blast kare (right-half $λ$ ), forward Euler uske saath blast karta hai, jabki backward Euler sirf modest steps ke liye blast karta hai aur phir, bahut bade steps ke liye, quietly explosion ko damp karta hai — decaying side pe unconditionally safe rehne ki honest cost. Sab possible fallers ke map mein, forward Euler ka safe zone ek chhota circle hai, jabki backward Euler ka safe zone decaying cheezoon ki poori duniya cover karta hai. Same accuracy ( $O (h)$ ), wildly better stability — aur yahi implicit jaane ka poora point hai.

Connections

Parent: Stiff equations & backward Euler
Forward Euler method
Region of absolute stability
A-stability and L-stability
Trapezoidal / Crank–Nicolson method
Newton's method for root finding
Runge–Kutta methods
Eigenvalues and time scales of linear systems