Stiff equations — implicit methods, backward Euler

4.8.26 · Maths › Numerical Methods

Stiff equation KYA hota hai?

Model test problem lete hain (jis lens se hum HAR method ko study karte hain):

$y^{'} = λ y, y (0) = y_{0}, λ < 0 (or Re λ < 0)$

True solution: $y (t) = y_{0} e^{λ t} \to 0$ . $λ$ jitna zyada negative, utna faster decay, utna stiffer problem.

Forward Euler KAISE fail karta hai (derivation + stability)

Forward Euler derive karo derivative ki definition se:

$y^{'} (t) \approx \frac{y _{n + 1} - y _{n}}{h} \Rightarrow \frac{y _{n + 1} - y _{n}}{h} = f (t_{n}, y_{n})$

$y_{n + 1} = y_{n} + h f (t_{n}, y_{n})$

$y^{'} = λ y$ par apply karo:

$y_{n + 1} = y_{n} + hλ y_{n} = (1 + hλ) y_{n} \Rightarrow y_{n} = (1 + hλ)^{n} y_{0}$

Agar $λ = - 1000$ ho, to $h < 0.002$ chahiye — hamesha, chahe transient khatam ho gayi ho. Disaster.

Backward Euler KAISE fix karta hai (derivation + stability)

Backward Euler derive karo $t_{n + 1}$ par backward difference use karke:

$y^{'} (t_{n + 1}) \approx \frac{y _{n + 1} - y _{n}}{h} = f (t_{n + 1}, y_{n + 1})$

$y_{n + 1} = y_{n} + h f (t_{n + 1}, y_{n + 1})$

Ye implicit hai: $y_{n + 1}$ dono sides par unknown hai. $y^{'} = λ y$ par apply karo:

$y_{n + 1} = y_{n} + hλ y_{n + 1} \Rightarrow (1 - hλ) y_{n + 1} = y_{n}$

$y_{n + 1} = \frac{1}{1 - hλ} y_{n} \Rightarrow y_{n} = (\frac{1}{1 - hλ})^{n} y_{0}$

Toh backward Euler decay karne wali problems ke liye unconditionally stable hai — $h$ accuracy ke liye choose karo, survival ke liye nahi.

Figure — Stiff equations — implicit methods, backward Euler

Implicit equation actually solve KAISE karte hain?

Linear $f$ ke liye sirf rearrange karo (jaise upar). Nonlinear $f$ ke liye, $y_{n + 1} = y_{n} + h f (t_{n + 1}, y_{n + 1})$ ek root-finding problem hai. Har step par Newton's method se solve karo:

$g (y) = y - y_{n} - h f (t_{n + 1}, y) = 0, y^{(k + 1)} = y^{(k)} - \frac{g ( y ^{(k)} )}{g ^{'} ( y ^{(k)} )}, g^{'} = 1 - h \frac{\partial f}{\partial y} .$

Worked examples

Common mistakes

Recall Feynman: 12-saal ke bacche ko explain karo

Socho ek jhula jo super fast shant ho jaata hai. Agar tum sirf dekhte ho aur andaaza lagaate ho ki ek moment baad kahan hoga (forward Euler), aur zyada der baad dekho, tera andaaza wildly overshoot kar jaata hai aur tum "gir jaate ho." Backward Euler zyada smart hai: tum poochte ho "mujhe kahaan hona chahiye taaki cheezein AGLE moment par shant rahein?" aur uske liye solve karte ho. Kyunki tum shant future ka aim karte ho, kabhi overshoot nahi hota — toh bade, aaram se steps le sakte ho aur phir bhi sahi nikalta hai.

Flashcards

Ek ODE stiff kyun hoti hai?

Uske solution components bahut alag time scales par hote hain, jo explicit methods ko stability ke liye tiny

h

use karne pe majboor karta hai chahe solution smooth ho.

Forward Euler formula (test problem

y^{'} = λ y

y_{n + 1} = (1 + hλ) y_{n}

Forward Euler stability condition?

∣1 + hλ ∣ < 1

; real

λ < 0

ke liye,

h < 2/∣ λ ∣

Backward Euler formula?

y_{n + 1} = y_{n} + h f (t_{n + 1}, y_{n + 1})

(implicit).

Backward Euler amplification on

y^{'} = λ y

y_{n + 1} = \frac{1}{1 - hλ} y_{n}

Backward Euler stability condition?

∣1/ (1 - hλ) ∣ < 1

, ye sab

h > 0

ke liye true hai jab

Re λ < 0

A-stable ka matlab kya hai?

Stability region poore left half-plane

Re (hλ) < 0

ko contain karta hai.

Backward Euler ki accuracy ka order?

First order,

O (h)

— forward Euler ke barabar.

Backward Euler ke andar Newton's method kyun use karte hain?

Nonlinear implicit equation solve karne ke liye bina us step-size restriction ke jo fixed-point iteration impose karta.

Implicit vs explicit mein defining difference kya hai?

Explicit

f

ko known

t_{n}

par use karta hai; implicit

f

ko unknown

t_{n + 1}

par use karta hai, toh

y_{n + 1}

solve karna padta hai.

Connections

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