Click/drag to add heat — watch it spread via du/dt = alpha * nabla^2 u
The heat equation du/dt = alpha * (d^2u/dx^2 + d^2u/dy^2) governs how temperature spreads.
Discretized as u[i][j] += alpha * dt * (u[i+1][j] + u[i-1][j] + u[i][j+1] + u[i][j-1] - 4*u[i][j]).
Stability requires alpha*dt/dx^2 < 0.25. Heat flows from hot to cold until equilibrium (maximum principle).
This same equation governs diffusion of chemicals, electric potential, and probability densities.