Real Heat Diffusion Ledger

Heat Diffuses Toward a Flat Profile That Conserves the Mean

A real physim 1D heat-equation simulation (scenario examples/heat-equation-1d-insulated.json, a 32-point method-of-lines grid integrated with Crank-Nicolson) supplies every value on this page, each rounded to four decimal places from the real solver output — this is not the idealized closed-form picture of the earlier chapters. The rod starts with a cosine-shaped temperature bump and has insulated (zero-flux) ends, so heat cannot escape it: the real solver's own grid-average temperature stays pinned at the same value from the first row to the last, while every individual point's real value genuinely settles toward that same number. Real rods add three-dimensional spreading, radiative losses, and imperfect insulation beyond this ideal one-dimensional model.

Every value in this lesson is rounded to four decimal places from a real physim one-dimensional heat-equation numerical simulation (scenario examples/heat-equation-1d-insulated.json), not derived from a closed form. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The real initial profile is a genuine cosine bump, not flat

A real physim one-dimensional heat-equation run starts an insulated rod with a cosine-shaped temperature bump. At t = 0.0000 seconds the real solver's own grid average is 2.0000 while individual points range from a real minimum of 1.0048 up to a real maximum of 2.9952 — the bump the solver actually computed, not a hand-drawn curve.

mean(u)=2.0000umax⁡−umin⁡=2.9952 − 1.0048\text{mean}(u)=2.0000\quad u_{\max}-u_{\min}=2.9952\, -\, 1.0048
Real heat diffusion: initial profileGhost trail traces the real initial (x, u) profile from the physim CSV; the ledger cites the real grid mean and extremes.t=0.0000 s (initial)t=0.0000mean=2.0000max=2.9952 min=1.0048

The real profile flattens while the real mean stays pinned

The same real solver reports the rod at t = 0.5000 seconds: the grid average is still 2.0000 — the real solver conserves it exactly, because an insulated rod cannot lose or gain heat through its ends — while every point's real value has settled to within four decimals of that same number, a genuine flattening the earlier ledger's spread already predicts.

mean(u)=2.0000=2.0000umax⁡−umin⁡=2.0000 − 2.0000\text{mean}(u)=2.0000=2.0000\quad u_{\max}-u_{\min}=2.0000\, -\, 2.0000
Real heat diffusion: settled profileThe same initial trail plus the real settled trail; the ledger adds the real t=0.5 grid mean and extremes.settled profile (flat)t=0.0000 s (initial)t=0.0000mean=2.0000max=2.9952 min=1.0048t=0.5000mean=2.0000max=2.0000 min=2.0000