I am looking for a semi-Lagrangian method for advection with a non divergence free velocity field.
The equation is
\begin{align} \frac{\partial C(x,t)}{\partial t} &= - \nabla \cdot (\vec{v}(x,t) C(x,t)) \\ C(x, 0) &= C_0(x) \end{align}
I've solved that using the formulation in this paper [1], which describes a quasi-monotone and conservative scheme, where they use a property of the Jacobian determinant to come up with
\begin{align} \frac{\partial J(x,t)}{\partial t} &= J \nabla \cdot \vec{v}(x,t) \\ D(x, 0) &= 1 \end{align} where $J$ is the determinant of the Jacobian.
Then they replace the first 2 equations with
\begin{align} \frac{D J(x,t)}{D t} &= 0 \\ C(x, 0) &= C_0(x) \end{align}
I have managed to implement it and it works fine. My worries is that the paper is a bit old (2002) and I would like to know if there is another way to achieve that with a semi-Lagrangian scheme? I think the main drawback of this approach is that conservation is achieved globally and not locally, and that the use of the Jacobian prevents the use of dimensional splitting.
I've not found easier alternatives in the literature I have read. I have read a lot about the combination with WENO/ENO scheme that seems to be promising, but I didn't find a case for non divergence free velocity field.
Is there some more up to date techniques than the one I used?