Numerical Methods · interactive worked example
The same circular fin from the steady-state lab, now heating up from room temperature over time. Step it forward explicitly and there's a hard speed limit on the time step — cross it, and the "solution" detonates into nonsense within a few dozen steps. Step it forward implicitly and there's no limit at all.
A transient extension of the Circular Fin worked example — same fin, now solved in time as well as space.
θ(x,t) — heating up from ambient toward the steady profile
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Explicit time-stepping computes tomorrow's temperature using only today's neighbors — fast, but it implicitly assumes the disturbance can't travel faster than information moves through the grid in one step. Make the time step too large relative to the grid spacing, and that assumption breaks: errors stop decaying and start amplifying each step instead, doubling and redoubling until the numbers overflow into nonsense. That's not the physics failing — the real fin doesn't explode — it's the numerical scheme failing, which is exactly why the stability limit has to be respected, not just hoped past.
Implicit time-stepping computes tomorrow's temperature using tomorrow's neighbors too — which means solving a small linear system (the same tridiagonal-matrix idea as the Gauss-Seidel lab) at every step instead of just evaluating a formula. That extra cost buys unconditional stability: any time step, however large, gives a bounded answer. Large steps still lose accuracy — you can step right past interesting transient behavior — but they never blow up the way explicit does.
Push the time-step ratio past 1.0 with Explicit selected and press Play — watch the curve start oscillating, then diverge entirely within a handful of steps. Switch to Implicit at the exact same ratio and watch it calmly settle onto the same steady profile from the Circular Fin lab instead.
EngineeringCandy · Same fin, same matrix-building ideas, now marching forward in time · break the time step on purpose