Question 1

What is the heat equation?

Accepted Answer

The heat equation (or diffusion equation) is a partial differential equation that describes how temperature evolves through a material over time: ?T/?t = a?�T, where T is temperature, t is time, and a is thermal diffusivity (a = k/(?c), with k = conductivity, ? = density, c = specific heat capacity). The Laplacian ?�T captures the second spatial derivative � heat flows from hot regions to cold, at a rate proportional to the local curvature of the temperature field.

Question 2

What is thermal diffusivity?

Accepted Answer

Thermal diffusivity a = k/(?c) measures how quickly temperature changes propagate through a material. High a (e.g. copper: 1.17 � 10?4 m�/s) means heat spreads rapidly; low a (e.g. wood: 8 � 10?8 m�/s) means slow penetration. It combines thermal conductivity k (how well heat is conducted), density ? (mass per volume), and specific heat capacity c (energy needed to raise temperature). Materials with high a make better heat sinks; low-a materials work well as insulators.

Question 3

What are Dirichlet and Neumann boundary conditions?

Accepted Answer

Dirichlet boundary conditions fix the temperature value at the boundary (T = T_boundary). Examples: a rod end held at a constant temperature, a wall touching an ice bath at 0 �C. Neumann boundary conditions fix the heat flux (the normal derivative ?T/?n = q). A zero-flux (insulated) boundary ?T/?n = 0 prevents heat from leaving � modelling a perfectly insulated surface. Specifying a non-zero flux models a heater or cooler imposing a known rate of energy transfer per unit area.

Question 4

What is the CFL stability condition for the heat equation?

Accepted Answer

The Courant-Friedrichs-Lewy (CFL) stability condition for the explicit finite-difference heat equation in 2D is r = a�?t/?x� = 0.25. If r exceeds this limit, round-off errors grow exponentially � the simulation 'explodes'. To maintain stability you must either reduce the time step ?t, increase the spatial resolution ?x, or reduce diffusivity a. The simulator enforces r = 0.24 at all times, automatically clamping any slider combination that would violate stability.

🧪 Heat Equation Simulator: Thermal Diffusion and Conduction

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