I have to numerically solve the equation \begin{equation*} \frac{\partial u}{\partial t} - D \Delta u + B \cdot \nabla u + c u = f \end{equation*} with non-homogeneous Neumann boundary conditions. Without the $c u$ term, this equation is the convection-diffusion equation. I have tried to figure out the physical significance of the extra $c u$ term, but I couldn't find anything. Do any of you know if this is a known equation that models some natural phenomenon?
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A term linear in $u$ is a birth/death term. The population grows or dies exponentially with $$\partial_t \log u(t) ==-c $$
if all other terms are discarded, that means in spatial equilibrium.
The Div term is an effect of a moving system of coordinates
$$ t-> t', x->x' -v t', \partial_t -> \partial_t' - v \partial_x $$
yielding (swap of a terms over the =,yields, sometimes, something new)
$$ \vec x ->vec x - \vec B t , \nabla .\ nabla -> \Delta - \Div \vec B $$
This is called the Galilileo group of transformations to an uniform moving system of space-time coordinates.
Even if it is not a group in the canonical sense.
Evidently the central definitions of the norm of the time derivative is changed; in analytical mechanics its called "kinetic energy" which depends on the speed of the spatial coordinate system.
Roland F
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