# Numerical methods for the continuity equation with Sobolev vector field

Consider the continuity equation $$\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,$$ with $$b \in L^1((0,T), W^{1,p}(\mathbb R^N))$$.

• Have any numerical methods been used to approximate the (unique renormalized) distributional solution of this problem? Can you point out any references on this topic?

• What if we replace $$W^{1,p}$$ with $$BV$$?

Clearly, if $$b$$ is smooth, the problem is much more classical.

I've asked a related question on MathOverflow and a specific implementation problem (with a specific $$b$$ of bounded variation) on Mathematica StackExchange.

• I don't know much about numerical methods in this area, but for theory I would take a look at the books by Lions. – Wolfgang Bangerth Apr 21 '19 at 23:41