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Has any work been done about numerical methods for the continuity equation $$ \partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N, $$ where $b \in L^1_tW^{1,p}_x$?

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  • $\begingroup$ @DavidKetcheson I wrote a short-hand for $L^1((0,T), W^{1,p}(\mathbb{R}^N))$. $\endgroup$
    – Riku
    Commented Apr 22, 2019 at 15:56
  • $\begingroup$ Can you give an example of a function $b(x,t)$ for which standard numerical methods don't work? $\endgroup$
    – David Ketcheson
    Commented Apr 22, 2019 at 16:29
  • $\begingroup$ @DavidKetcheson I don't know what works (that's implicit in the question), but my understanding is that you need to assume $b$ smooth to get convergence estimates for numerical methods for the continuity equation. $\endgroup$
    – Riku
    Commented Apr 22, 2019 at 16:34

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While the standard numerical methods (at least some of them) work even in the Sobolev regularity setting, the analysis of convergence is far from being trivial (due to non-smoothness of $b$). It is addressed for instance in the papers

  1. A. Schlichting, C. Seis: Convergence rates for upwind schemes with rough coefficients SIAM J. Numer. Anal., 55(2), 2017.

  2. A. Schlichting, C. Seis: Analysis of the implicit upwind finite volume scheme with rough coefficients, Numer. Math. 139(1), 2018.

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