I am not sure has the following observation been made:

When the Jacobi Method is used to approximate the solution of Laplace’s Equation, if the initial temperature distribution is given by , then the iterations are also approximations to the solution, , of the Heat Equation, assuming the initial temperature distribution is .

I first considered such a thought while looking at approximations to the solution of Laplace’s Equation on a thin plate. The way I implemented the approximations was I wrote the iterations onto an Excel worksheet, and also included conditional formatting to represent the areas of hotter and colder, and the following kind of output was produced:

Let me say before I go on that this was an implementation of the Gauss-Seidel Method rather than the Jacobi Method, and furthermore the stopping rule used was the rather crude .

However, do not the iterations resemble the flow of heat from the heat source on the bottom through the plate? The aim of this post is to investigate this further. All boundaries will be assumed uninsulated to ease analysis.

## Discretisation

Consider a thin rod of length . If we *mesh* the rod into pieces of equal length , we have *discretised *the rod, into segments of length , together with ‘nodes’ .

Suppose are interested in the temperature of the rod at a point , . We can instead consider a sampling of , at the points :

.

Similarly we can *mesh *a plate of dimensions into an rectangular grid, with each rectangle of area , where and , together with nodes , and we can study the temperature of the plate at a point by sampling at the points :

.

We can also *mesh *a box of dimension into an 3D grid, with each rectangular box of volume , where , , and , together with nodes , and we can study the temperature of the box at the point by sampling at the points :

.

## Finite Differences

How the temperature evolves is given by *partial differential equations*, expressing relationships between and its rates of change.

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