shifting mass along a vector field

I have a positive matrix $$\rho \in \mathbb{R}^{n,n}_+$$ as a discrete probability density. Furthermore, I have a tensor $$u \in \mathbb{R}^{n,n,2}$$ that is constant in time and acts as a vector field. I want to transport the mass of $$\rho$$ along the discrete vector field $$u$$ with numpy. But I am not sure how to do this or where to start.

Edit: I guess, that the convection equation with discontinuous Galerkin method might work.