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: