NOTE: I asked this on Mathematics Stack Exchange and there were no answers. So, I thought I might try here.
Upwind schemes like the classic "upstream" scheme, can be used to solve, for example, the advection equation:
$$ \dfrac{\partial \psi}{\partial t} + \dfrac{\partial }{\partial x}(u\psi) = 0 \qquad (1) $$
and this makes sense, because the scalar $\psi$ is being transported by the wind speed $u$.
My question is, what is the extent to which these schemes could be used? Could they be used to solve all partial differential equations? Does it make sense? For example you can find in the literature attempts to solve the diffusion equation with an upwind method. And if we were to use them to solve an equation, for example, of the form,
$$ \dfrac{\partial \psi}{\partial t} + a(x,t)\dfrac{\partial \psi}{\partial x} = 0\qquad (2) $$
should we use a virtual wind speed, i.e.
$$ u=\dfrac{1}{\psi} \int a \dfrac{\partial \psi}{\partial x}dx $$
to determine the stencils?