2D Diffusion Equation

Suppose we are solving the diffusion equation

Constant diffusion

Assume everywhere.
Using the 5-point method, we get:

We can show that and
where is Kronecker Product and:

Then , so they commute.
Using Splitting Numerical Schemes we can write:

Split Crank-Nicolson now uses

to approximate the exponential

where the inverse is calculated in due to its nice structure (block diagonal with tridiagonal blocks).
Note that in we first need to appropriately permute the rows and columns to make it nice.
The stability is verified:

Variable diffusion

First replace all space derivatives by

to find the ODE system:

Then and do not necessarily commute
so by Product of matrix exponentials:

has an error
We can find a better approximation, for example: