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.

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

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