  This manuscript contains some thoughts on the discretization of the classical
heat equation. Namely, we discuss the advantages and disadvantages of explicit
and implicit schemes. Then, we show how to overcome some disadvantages while
preserving some advantages. However, since there is no free lunch, there is a
price to pay for any improvement in the numerical scheme. This price will be
thoroughly discussed below. In particular, we like explicit discretizations for
the ease of their implementation even for nonlinear problems. Unfortunately,
when these schemes are applied to parabolic equations, severe stability limits
appear for the time step magnitude making the explicit simulations
prohibitively expensive. Implicit schemes remove the stability limit, but each
time step requires now the solution of linear (at best) or even nonlinear
systems of equations. However, there exists a number of tricks to overcome (or
at least to relax) severe stability limitations of explicit schemes without
going into the trouble of fully implicit ones. The purpose of this manuscript
is just to inform the readers about these alternative techniques to extend the
stability limits. It was not written for classical scientific publication
purposes.
