Numerical Methods · Design background
The same fin, the same spatial discretization as before — the only new ingredient is time, and time is where a finite-difference scheme can quietly stop being trustworthy.
The steady fin answered "what does the temperature profile settle into?" This lab answers "how does it get there?" — by putting the time derivative back in:
$\alpha$, the thermal diffusivity, sets how fast a thermal disturbance spreads — the same quantity from the Transient Conduction lab, just now applied pointwise along the fin instead of lumped into one number for the whole body.
The most direct way to step forward: estimate tomorrow's temperature at each point using only today's values at that point and its neighbors.
This is just arithmetic — no linear system to solve, each step is a single pass over the grid. The catch is hiding in that $(1-2\mathrm{Fo}-\alpha m^2\Delta t)$ coefficient: if it goes negative, the scheme starts weighting yesterday's value by a negative number, which means any tiny error — even ordinary floating-point rounding — gets amplified instead of damped at every step. Demand that coefficient stay non-negative and you get the stability limit:
Once $\Delta t$ exceeds that limit, the error doesn't grow gently — it grows geometrically, flipping sign each step (you can see this as the curve starting to oscillate before it visibly diverges). Within a few dozen steps, ordinary double-precision floating point numbers overflow into values with no physical meaning at all. This is purely a property of the numerical scheme, not the physics: a real copper fin obviously doesn't develop a million-degree oscillation. The lesson is that "the computer gave an answer" and "the answer means something" are two different claims, and a stability analysis is what tells you whether you're allowed to believe the first implies the second.
Try the "Blow it up" preset with Explicit selected, then switch to Implicit at the identical time-step ratio — same equation, same grid, wildly different fate. Open the lab →
Implicit time-stepping writes tomorrow's value in terms of tomorrow's neighbors too, which means every node's new value depends on every other new value — solved all at once as a small tridiagonal linear system (the Gauss-Seidel lab solves general dense or sparse systems the same way, just iteratively instead of by direct elimination here). That coupling is exactly what buys unconditional stability: there's no analogous "coefficient goes negative" failure mode, so any $\Delta t$ gives a bounded, physically sensible answer. The tradeoff is pure efficiency, not correctness — a too-large implicit step can step right past real transient behavior you wanted to see, but it will never diverge the way explicit does.
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