Non-conservative advective term in a finite volume scheme

Question

I am interested in solving this set of nonlinear couples advection-diffusion equations using a finite volume scheme: $$ \frac{\partial f(x,y)}{\partial t}=-(\boldsymbol{u}+\nabla\eta)\cdot\nabla f +\nabla\cdot(\eta\nabla f)+s_f(g), $$ $$ \frac{\partial g(x,y)}{\partial t}=-\nabla\cdot(\boldsymbol{u}g)+\nabla\cdot(\eta\nabla g)+\nabla\cdot(\boldsymbol{s_g}(f))-l_g(g). $$ As you can see this set of equations lends itself well to the use of a finite volume scheme, except for the advective term in the first equation, which is non-conservative. So my question is this: I assume we must still take particular care for the discretisation of that term?

I must say that I am a bit confused. I'm reading Chapter 4 of Ferziger et al. 2002 on the finite volume method. They mention that upwind interpolation is the FVM equivalent of the FDM upwind differencing. Then they go on about linear interpolation being the simplest and most widely used interpolation method for the face values, which is equivalent to central difference. But central difference is supposed to be unstable for the advective term. I assume this is not the case here because we used Gauss' theorem to get rid of the derivatives? Then for the non-conservative advective term I have to use another approach? What would be the best second-order method for a code that will be fully explicit in time?

EDIT: Actually, there is something I forgot to mention that might be important. $f$ is essentially the poloidal vector potential (there is only one nonzero component) of one component of the magnetic field. The other component is $g$. This decomposition is possible because of symmetries. What this means, is that lines of constant $f$ represent the magnetic field lines of that magnetic field component. So advection only works perpendicularly to the direction given by the magnetic field lines, or: $$ -(\mathbf{u}+\nabla\eta)\cdot\nabla f=-(\mathbf{u}_{\perp}+\nabla_{\perp}\eta)\cdot\nabla_{\perp} f=-\nabla_{\perp}\cdot((\mathbf{u}_{\perp}+\nabla_{\perp}\eta)f)+f\nabla_{\perp}\cdot(\mathbf{u}_{\perp}+\nabla_{\perp}\eta). $$ Not sure of much that helps though. But I have a feeling that since $f$ represents magnetic field lines, it should somehow be conservative...

I also rewrote the first equation more accurately.

Answer 1

There is a methology for non-conservative products called path-conservative schemes, which might useful for you. The method can be applied to systems of the form

\begin{align} \frac{\partial \mathbf{Q}}{\partial t} &+ \nabla \cdot \mathbf{f}(\mathbf{Q})+\mathbf{B(\mathbf{Q})}\nabla\mathbf{Q}=\mathbf{S}(\mathbf{Q}),\\ \frac{\partial \mathbf{Q}}{\partial t} &+\hspace{1cm}\mathbf{A(\mathbf{Q})}\cdot\nabla \mathbf{Q}\hspace{0.81cm} = \mathbf{S}(\mathbf{Q}) , \quad \text{quasi-linear form} \\ \text{with} &\quad\mathbf{A(\mathbf{Q})}=\frac{\partial \mathbf{f}(\mathbf{Q})}{\partial \mathbf{Q}} +\mathbf{B(\mathbf{Q})}. \end{align}

Here $\mathbf{Q}$ is the conservative state vector, $\mathbf{f}$ is the conservative flux vector and $\mathbf{S}$ is a source term. The third term in the first line is the non-conservative product. In case of

\begin{align} \mathbf{B}(\mathbf{Q})=0, \end{align}

you recover the well known conservation law. You may read some papers of LeFloch, Parès or Castro. [1],[2]

Answer 2

Just convert it into conservative form plus a reaction term: $$ \mathbf u \cdot \nabla f = \nabla \cdot (\mathbf u f) - (\nabla \cdot \mathbf u) f. $$