Stiff equations — implicit methods, backward Euler: Foundations

This is the Foundations child of the stiff-equations topic. We assume you have seen none of the notation. We earn every symbol before using it.

1. What is a function of time, $y (t)$ ?

Picture a single number that changes as time passes: the temperature of a cooling cup of coffee, the height of a bouncing-then-settling swing, the charge left on a battery.

Figure — Stiff equations — implicit methods, backward Euler

Look at the blue curve above: at each moment $t$ (a spot on the bottom axis) the curve is at some height. That height is the single number $y (t)$ . The entire blue curve is the function; one dot on it is one value $y (t)$ . Reading the curve left-to-right is watching the system evolve.

Why the topic needs this: everything in stiff equations is about how such a curve decays toward zero and how well a computer can trace it.

2. The derivative $y^{'}$ — the slope of the curve

A computer cannot store the whole curve; it can only know the value now and a rule for how fast it is changing. That rate of change is the derivative.

The green line is the tangent — the straight line that touches the curve and matches its slope at that one spot. Its steepness is the number $y^{'}$ .

Why "derivative" and not some other tool? Because a differential equation is a machine that hands you the slope at every point and asks you to reconstruct the curve. Slope is exactly the language it speaks.

3. The differential equation $y^{'} = f (t, y)$ and its starting point $y_{0}$

Read it aloud: "the rate of change of $y$ equals some formula involving the current time and the current $y$ ."

But a slope-rule alone is not enough. Knowing the slope everywhere tells you the shape of the curve, not where it starts. Infinitely many parallel curves have the same slope-rule; we must pin down one.

Why the topic needs this: the whole subject studies one master IVP, $y^{'} = λ y$ with $y (0) = y_{0}$ , because it is the simplest decay-rule and every method's behaviour shows up cleanly in it.

4. The exponential $e^{λ t}$ — the shape of pure decay/growth

But $e^{λ t}$ is not the only solution of $y^{'} = λ y$ — any constant multiple works too, because scaling a curve scales its slope by the same factor. The starting value $y_{0}$ from Section 3 fixes which multiple:

Two decaying curves: $λ = - 1$ (gentle, yellow) and $λ = - 100$ (a cliff, red). Both head to zero, but the red one is essentially gone almost instantly. A stiff problem contains both shapes at once — and that clash of speeds is the whole difficulty.

5. Complex $λ$ and " $Re λ$ "

Sometimes $λ$ is a complex number (it has a "real part" and an "imaginary part") because oscillations hide inside it. You do not need complex analysis here — only one word.

Why the topic needs it: real systems (springs, circuits) decay and oscillate. Writing $Re λ < 0$ lets one clean inequality cover them all.

6. The discrete world: $h$ , $t_{n}$ , $y_{n}$

A computer cannot handle a smooth continuum. It jumps forward in fixed hops.

The smooth blue curve is the truth; the yellow dots are the computer's samples $y_{0}, y_{1}, y_{2}, \dots$ spaced $h$ apart. The very first dot $y_{0}$ is handed to us by the initial condition; numerical methods are recipes for getting the next dot from the current one.

7. The approximation $y^{'} (t_{n}) \approx \frac{y _{n + 1} - y _{n}}{h}$

Here is where "slope" meets "hops". A slope is rise over run. Over one hop, the run is $h$ and the rise is the change in $y$ .

This is the bridge from the exact world (Section 2) to the computable world (Section 6). Which dot's slope this rise-over-run represents — the old dot $t_{n}$ or the new dot $t_{n + 1}$ — is the single choice that splits forward Euler (explicit) from backward Euler (implicit). That choice is the heart of the parent topic; you now have every symbol to understand it.

8. Absolute value $∣ \cdot ∣$ — the size that decides stability

Why the topic needs it: a method is stable when, step after step, it shrinks the value toward zero rather than blowing it up. "Does it shrink?" is exactly "is the multiplier's magnitude below $1$ ?" — written $∣ multiplier ∣ < 1$ . Sign doesn't matter (a value could flip sign each step and still shrink); only size does. That is why absolute value, not the plain number, rules stability.

9. Prerequisite map

Each foundation feeds the next: time and value make a curve, the curve has a slope, the slope-rule plus a starting value $y_{0}$ is an IVP, the IVP's answer is a scaled exponential, mixing fast and slow exponentials makes stiffness, and the finite-difference slope plus the size-test decide whether a method survives.

Equipment checklist

Self-test: can you answer each before revealing it?

What does the single value

y (t)

mean, versus the function

y (\cdot)

y (t)

is one number — the height at that one time

t

; the function

y (\cdot)

is the whole curve, every value at every time.

What is the derivative

y^{'}

The slope/steepness of the curve at a point — how fast

y

changes per tiny bit of time.

What does

y^{'} = f (t, y)

say?

A rule giving the current slope from the current time and current value;

f

is the slope-rule.

What extra ingredient turns a slope-rule into an IVP, and why?

The initial condition

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

; it pins down where the curve starts, selecting one curve from the infinite family with that slope-rule.

What is

y_{0}

The starting value of

y

at the start time

t_{0}

— the one number we know exactly, and the first sample.

What curve solves

y^{'} = λ y

with

y (0) = y_{0}

y (t) = y_{0} e^{λ t}

, whose slope is always

λ

times its height and which starts at

y_{0}

What does

λ < 0

do to the solution?

Makes it decay toward zero — more negative means faster decay.

What does

Re λ < 0

mean?

The real part of

λ

is negative, i.e. the system decays (allowing hidden oscillation).

What are

h

t_{n}

, and

y_{n}

Step size, the time

nh

after

n

hops, and the computed estimate of

y

there.

How do we approximate

y^{'} (t_{n})

with samples?

By the finite difference

(y_{n + 1} - y_{n}) / h

— rise over run between the dots at

t_{n}

and

t_{n + 1}

Why does stability use

∣ \cdot ∣

and not the plain value?

Because stability asks "does the value shrink each step?", which depends on the multiplier's size, not its sign.

Difference between

y_{n}

and

y (t_{n})

y_{n}

is the computed dot (may err or explode);

y (t_{n})

is the exact value; their gap is the error.