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Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations

  • * Corresponding author: Federica Bubba

    * Corresponding author: Federica Bubba 
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  • The parabolic-elliptic Keller-Segel equation with sensitivity saturation, because of its pattern formation ability, is a challenge for numerical simulations. We provide two finite-volume schemes that are shown to preserve, at the discrete level, the fundamental properties of the solutions, namely energy dissipation, steady states, positivity and conservation of total mass. These requirements happen to be critical when it comes to distinguishing between discrete steady states, Turing unstable transient states, numerical artifacts or approximate steady states as obtained by a simple upwind approach.

    These schemes are obtained either by following closely the gradient flow structure or by a proper exponential rewriting inspired by the Scharfetter-Gummel discretization. An interesting fact is that upwind is also necessary for all the expected properties to be preserved at the semi-discrete level. These schemes are extended to the fully discrete level and this leads us to tune precisely the terms according to explicit or implicit discretizations. Using some appropriate monotonicity properties (reminiscent of the maximum principle), we prove well-posedness for the scheme as well as all the other requirements. Numerical implementations and simulations illustrate the respective advantages of the three methods we compare.

    Mathematics Subject Classification: Primary: 35K55, 35Q84, 65M08, 65M22, 92C17.


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  • Figure 1.  Left: Comparison of solutions of the Scharfetter-Gummel (red line) and upwind (blue, dashed line) schemes at time $t = 100$ with the exact stationary solution (black line) for the linear Fokker-Planck equation with $\varphi (u) = u$. We used $I = 100$ and $\Delta t = 0.01$. Right: normalized $L^\infty$ variation for the two schemes

    Figure 2.  Evolution in time of solutions to (25) in the logistic case $ \varphi (u) = u (1-u) $ with $ \chi / D = 40 $. We solved the equation with the Scharfetter-Gummel (red line) and the gradient flow scheme (black dashed line) with $ I = 100 $ and $ \Delta t = 1 $. There is no major difference between the solutions given by the two schemes

    Figure 3.  Evolution in time of solutions to 25 in the logistic case $ \varphi (u) = u (1-u) $ with $ \chi / D = 40 $. We solved the equation with the Scharfetter-Gummel (red line) and the upwind scheme (blue, dashed line) with $ I = 100 $ and $ \Delta t = 1 $

    Figure 4.  Stationary profiles and dynamics. (A), (B) Comparison of the stationary profiles of solutions to the Scharfetter-Gummel (red line) and the upwind (blue, dashed line) schemes at $t = 50$ and $t = 9000$. (C) Normalized $L^\infty$ variation for the three schemes

    Figure 5.  Evolution in time of solutions to 25 in the exponential case $ \varphi (u) = u e^{-u} $ with $ \chi / D = 24 $. We solved the equation with the Scharfetter-Gummel (red line) and the gradient flow schemes (black, dashed line) with $ I = 100 $ and $ \Delta t = 1 $. As for the logistic model, the two schemes give the same solution

    Figure 6.  Stationary profiles and dynamics. (A), (B)Comparison of the stationary profiles obtained with the Scharfetter-Gummel (red line) and the upwind scheme (blue, dashed line) at $t = 50$ (left) and $t = 200$. (C) Normalized $L^\infty$ variation for the three schemes

    Figure 7.  Evolution in time of solutions to 25 in the exponential case $ \varphi (u) = u e^{-u} $ with $ \chi / D = 24 $. We compare the solutions of the Scharfetter-Gummel (red line) and the upwind schemes (blue, dashed line) obtained with $ I = 100 $ and $ \Delta t = 1 $ for different times

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