# American Institute of Mathematical Sciences

March  2019, 14(1): 23-41. doi: 10.3934/nhm.2019002

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

 Sorbonne Université, CNRS, Université Paris-Diderot SPC, Inria, Laboratoire Jacques-Louis Lions, 4, pl. Jussieu 75005, Paris, France

* Corresponding author: Federica Bubba

Received  April 2018 Revised  June 2018 Published  January 2019

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.

Citation: Luis Almeida, Federica Bubba, Benoît Perthame, Camille Pouchol. Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations. Networks & Heterogeneous Media, 2019, 14 (1) : 23-41. doi: 10.3934/nhm.2019002
##### References:

show all references

##### References:
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
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
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 $\Delta t = 1$
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
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
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
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
 [1] Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure & Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339 [2] Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243 [3] Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768 [4] Jinhuan Wang, Li Chen, Liang Hong. Parabolic elliptic type Keller-Segel system on the whole space case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1061-1084. doi: 10.3934/dcds.2016.36.1061 [5] Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597 [6] Hongyun Peng, Zhi-An Wang, Kun Zhao, Changjiang Zhu. Boundary layers and stabilization of the singular Keller-Segel system. Kinetic & Related Models, 2018, 11 (5) : 1085-1123. doi: 10.3934/krm.2018042 [7] Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231 [8] Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078 [9] Jaewook Ahn, Kyungkeun Kang. On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5165-5179. doi: 10.3934/dcds.2014.34.5165 [10] Kenneth H. Karlsen, Süleyman Ulusoy. On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion. Networks & Heterogeneous Media, 2016, 11 (1) : 181-201. doi: 10.3934/nhm.2016.11.181 [11] Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317 [12] Luca Battaglia. A general existence result for stationary solutions to the Keller-Segel system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 905-926. doi: 10.3934/dcds.2019038 [13] Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler. Mathematical methods in systems biology. Mathematical Biosciences & Engineering, 2016, 13 (6) : i-ii. doi: 10.3934/mbe.201606i [14] Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks & Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401 [15] Sachiko Ishida, Yusuke Maeda, Tomomi Yokota. Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2537-2568. doi: 10.3934/dcdsb.2013.18.2537 [16] Jitraj Saha, Nilima Das, Jitendra Kumar, Andreas Bück. Numerical solutions for multidimensional fragmentation problems using finite volume methods. Kinetic & Related Models, 2019, 12 (1) : 79-103. doi: 10.3934/krm.2019004 [17] Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569 [18] Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks & Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689 [19] Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049 [20] José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini, Gershon Wolansky. Numerical study of a particle method for gradient flows. Kinetic & Related Models, 2017, 10 (3) : 613-641. doi: 10.3934/krm.2017025

2018 Impact Factor: 0.871