Heat Conduction

It is the 2D heat (or diffusion) equation, written ∂T/∂t = α∇²T, where T is temperature, t is time, α is the thermal diffusivity and ∇² is the Laplacian. It states that the rate of temperature change at a point is proportional to how much that point differs from the average of its surroundings, which makes heat flow from hot regions to cold ones.

Forward-Time Central-Space replaces the derivatives with finite differences on the grid. Each cell's next temperature equals its current value plus α times the sum of its four neighbours minus four times itself. Because the new field depends only on the old field, every cell can be updated independently in a single explicit pass, which is fast but limited by a stability condition.

Diffusivity α scales how strongly each cell pulls toward the average of its neighbours, so higher α makes heat spread faster and reach equilibrium sooner. Steps / frame sets how many FTCS updates run before each redraw, effectively a time-speed control: more steps per frame advances simulated time faster but does not change the underlying physics.

The model is a faithful discretisation of the ideal heat equation for an isotropic, homogeneous medium with fixed-temperature (Dirichlet) edges at 0 °C. It is a teaching tool, so it uses dimensionless grid units rather than real material constants, and it ignores convection, radiation and temperature-dependent properties. The qualitative behaviour of conductive heat flow is, however, captured correctly.

The four edges of the grid are clamped to 0 °C, so they continually drain thermal energy out of the domain. With no internal heat source sustaining it, conduction smooths every gradient and energy leaks through these fixed boundaries until the whole field relaxes to the boundary value of zero. Repainting hot or cold spots injects fresh energy and restarts the diffusion.