Let us discretise our full time dependent Diffusion Equation
both in space and in time. One of the ways to discretise

is

This is called the

But this is not the only way. Another way, which seems just as good,
but is, in fact, a lot better, is

This is called the

- Explicit schemes are very easy to program. In case of the Diffusion Equation they can be all reduced to the Jacobi scheme.
- Explicit schemes can take a very long time to converge.
- The (Richard) Courant - (Hans) Lewy stability condition very severely restricts the length of the time step for parabolic problems.
- The Jacobi scheme corresponds to the longest practical time step that is still compatible with the Courant-Lewy condition.
- The implicit scheme is
*unconditionally*stable, sic! But*extremely*hard to program. It results in a very large linear system. For example, for a grid you must solve 40,000 simultaneous linear equations for 40,000 unknowns. The equation matrix alone, if you were so insane as to try to code it literally, would have entries. Luckily the matrix is*sparse*and the problem is solvable. - There is an even better way to tackle the Diffusion
Equation, which is a lot easier to program than the implicit
scheme, but unlike the explicit scheme, solves the
whole problem without iterations in one swoop: the
*spectral*method, i.e., a method that is based on Fourier Transform.