5.2. Convection-diffusion equation

A very important equation for transport phenomena is the convection diffusion equation

\[\partial_t c + \nabla\cdot\left(c\vec{u}\right)=\nabla\cdot\left(D\nabla c\right) \,.\]

Here \(c\) can be understood as e.g. some concentration field or a temperature field, \(\vec{u}\) is the advecting velocity field and \(D\) is a diffusion coefficient. After multiplication with a test function \(\phi\), we can have two different weak formulations, depending on whether the advection term is included in the partial integration or not, namely:

(5.3)\[\left(\partial_t c,\phi\right) + \left(\nabla\cdot\left(c\vec{u}\right),\phi\right)+\left(D\nabla c,\nabla \phi\right) +\left\langle -D\nabla c\cdot \vec{n},\phi\right\rangle=0\]

and

(5.4)\[\left(\partial_t c,\phi\right) - \left(c\vec{u},\nabla\phi\right)+\left(D\nabla c,\nabla \phi\right) +\left\langle \left(\vec{u}c-D\nabla c\right)\cdot \vec{n},\phi\right\rangle=0\,.\]

Note that, if the velocity \(\vec{u}\) is divergence free (incompressible), the second term in (5.3) reads \((\vec{u}\cdot\nabla c,\phi)\). The first observation between both variants is that the Neumann term \(\langle \cdot,\cdot \rangle\) is different. In (5.3), we impose pure diffusive fluxes as Neumann conditions, whereas in (5.4) total fluxes, i.e. the sum of advection and diffusion, are imposed by Neumann conditions.

The second observation is more severe: The advection terms are not symmetric with respect to the spatial derivative order. While the time derivative has zeroth order spatial derivatives on both \(c\) and \(\phi\) and the diffusion term has both first order spatial derivatives, i.e. \(\nabla c\) and \(\nabla \phi\), the advection term is mixed. Either the field \(c\) or the test function \(\phi\) is derived. This asymmetry leads to considerable complications if the equation is advection-dominated.