Heat Diffusion

Heat flows from hot to cold. A point warms up when its neighbors are hotter than it on average, so the whole plate slowly evens out.

How it works

The heat equation

In one sentence

Temperature $u$ in a material of diffusivity $\alpha$ satisfies $u_t = \alpha\nabla^2 u$ : each point warms or cools in proportion to how much its surroundings differ from it.

General formula

{\htmlData{sym=0}{\frac{\partial u}{\partial t}}} = {\htmlData{sym=2}{\alpha}}\,{\htmlData{sym=1}{\nabla^2 u}}

$\nabla^2 u$ is the average of the neighbors minus the point itself. Heat flows from hot to cold, so every bump flattens.

What the symbols mean

$\frac{\partial u}{\partial t}$
Heating rate
u is the plate's temperature at each point. This is how fast it is changing there.
$\nabla^2 u$
Laplacian
Average of the neighbors minus this point. Positive when the surroundings are warmer, so the point heats up.
$\alpha$
Thermal diffusivity
How quickly the material spreads heat. Copper is high; wood is low.

Temperature changes at a rate proportional to the Laplacian: the gap between a point and its neighbors' average.
A computer updates every grid point with $u \leftarrow u + r\,(\text{neighbors} - 4u)$ , where $r = \alpha\,\Delta t/\Delta x^2$ .
If $r > \tfrac14$ the simulation explodes. Stability is a math constraint, not a coding bug.
Left alone, the plate settles into a steady state that solves Laplace's equation $\nabla^2 u = 0$ .

Learn the math

Higher Order Partial DerivativesCalc III · Lesson 21, Partial Derivatives The Divergence TheoremCalc III · Lesson 50, Surface Integrals Fourier Sine SeriesDiff Eq · Lesson 44, Boundary Value Problems and Fourier Series What Is a PDE?PDE · Lesson 1, Foundations The Heat EquationPDE · Lesson 4, Foundations Solving the Heat EquationPDE · Lesson 7, Separation of Variables Heat in a Rectangular PlatePDE · Lesson 15, Beyond the Basics Finite Difference MethodsPDE · Lesson 18, Beyond the Basics

Learn the math

Higher Order Partial DerivativesCalc III · Lesson 21, Partial Derivatives

The Divergence TheoremCalc III · Lesson 50, Surface Integrals

Fourier Sine SeriesDiff Eq · Lesson 44, Boundary Value Problems and Fourier Series

What Is a PDE?PDE · Lesson 1, Foundations

The Heat EquationPDE · Lesson 4, Foundations

Solving the Heat EquationPDE · Lesson 7, Separation of Variables

Heat in a Rectangular PlatePDE · Lesson 15, Beyond the Basics

Finite Difference MethodsPDE · Lesson 18, Beyond the Basics