Heat Equation Simulator
The heat equation uₜ = α uₓₓ governs how temperature diffuses: each point relaxes toward the average of its neighbours. This simulator evolves a 1D rod live from initial conditions you choose — a hot spot, a sharp step, two competing spots or random noise — with the ends held cold. Jagged features smooth out almost instantly while broad features decay slowly, showing why diffusion is nature’s low-pass filter, and the profile’s march toward flat equilibrium is drawn as a glowing temperature map.
Runs 100% in your browser — simulations are computed locally on your device.
Read the full guide to this tool
Notes
- Explicit finite differences: u ← u + r(u₊ + u₋ − 2u), stable only for r ≤ ½.
- High spatial frequencies decay fastest — mode k dies like e^(−αk²t), which is why sharp edges blur first.
- With fixed cold ends every initial condition eventually flattens to zero — equilibrium is total blandness.
- Runs 100% in your browser — simulations are computed locally on your device.
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