The diffusion equation,
for 
is parabolic since the discriminant,
.
The diffusion equation arises in diverse
settings, but most often in connection with a transient flow
problem in which the flow is down gradient of some state
variable u. In the setting of heat flow, the diffusion
equation (sometimes called the heat equation) could be used
to model a thermodynamics problem in which transient heat
flow is occurring in one space dimension. Similar to the
elliptic case, parabolic equations generally have very
smooth solutions. However, parabolic equations often
exhibit solutions with evolving regions of high gradient.
Most numerical methods for dealing with parabolic equations
involve approximating the solution at successive time
steps, with each approximation requiring the solution of a
system of linear equations. For these types of computational
problems, it is often useful to employ some matrix
factorization method in conjunction with a dynamic gridding
algorithm. Multispace generalizations of this example
problem can be solved efficiently on vector architectures
using ADI methods or on parallel architectures with some
divide and conquer strategies.
