# 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:
 [1] N. J. Armstrong, K. J. Painter and J. A. Sherratt, A continuum approach to modelling cell-cell adhesion, Journal of Theoretical Biology, 243 (2006), 98-113. doi: 10.1016/j.jtbi.2006.05.030. [2] J. Barré, J. A. Carrillo, P. Degond, D. Peurichard and E. Zatorska, Particle interactions mediated by dynamical networks: Assessment of macroscopic descriptions, Journal of Nonlinear Science, 28 (2018), 235-268. doi: 10.1007/s00332-017-9408-z. [3] A. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710. doi: 10.1088/0951-7715/22/3/009. [4] M. Bessemoulin-Chatard and F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations, SIAM Journal on Scientific Computing, 34 (2012), 559-583. doi: 10.1137/110853807. [5] A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM Journal on Numerical Analysis, 46 (2008), 691-721. doi: 10.1137/070683337. [6] A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electronic Journal of Differential Equations, 2006 (2006), 1-32. [7] F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-balanced Schemes for Sources, Birkhäuser, 2004. doi: 10.1007/b93802. [8] J. A. Carrillo, A. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233-258. doi: 10.4208/cicp.160214.010814a. [9] J. A. Carrillo, N. Kolbe and M. Lukacova-Medvidova, A hybrid mass-transport finite element method for Keller-Segel type systems, preprint, arXiv: 1709.07394, (2017). [10] J. A. Carrillo, Y. Huang and M. Schmidtchen, Zoology of a non-local cross-diffusion model for two species, SIAM Journal on Applied Mathematics, 78 (2018), 1078-1104. doi: 10.1137/17M1128782. [11] T. Cieślak and C. Morale-Rodrigo, Quasilinear non-uniformly parabolic-elliptic system modelling chemotaxis with volume filling effect. Existence and uniqueness of global-in-time solutions, Topological Methods in Nonlinear Analysis, 29 (2007), 361-381. [12] J. Dolbeault, P. Markowich and G. Jankowiak, Stationary solutions of Keller-Segel type crowd motion and herding models: Multiplicity and dynamical stability, Mathematics and Mechanics of Complex Systems, 3 (2015), 211-242. doi: 10.2140/memocs.2015.3.211. [13] R. Eymard, T. Gallouét and R. Herbin, Finite volume methods, Handbook of Numerical Analysis, 7 (2000), 713-1020. [14] T. Hillen and K. J. Painter, Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Applied Mathematics Quarterly, 10 (2002), 501-543. [15] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [16] T. Hillen, K. J. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete and Continuous Dynamical Systems Series B, 7 (2007), 125-144. doi: 10.3934/dcdsb.2007.7.125. [17] S. Jin and B. Yan, A class of asymptotic-preserving schemes for the Fokker-Planck-Landau equation, Journal of Computational Physics, 230 (2011), 6420-6437. doi: 10.1016/j.jcp.2011.04.002. [18] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [19] J. R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253. [20] J. Liu, L. Wang and Z. Zhou, Positivity-preserving and asymptotic preserving method for 2D Keller-Segel equations, Mathematics of Computation, 87 (2018), 1165-1189. doi: 10.1090/mcom/3250. [21] J. D. Murray, Mathematical Biology, vol. I: An Introduction, Springer, New York, 2002. [22] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Advances in Mathematical Sciences and Applications, 5 (1995), 581-601. [23] B. Perthame, Transport Equations in Biology, Birkhäuser Verlag, Basel, 2007. [24] A. B. Potapov and T. Hillen, Metastability in chemotaxis models, Journal of Dynamics and Differential Equations, 17 (2005), 293-330. doi: 10.1007/s10884-005-2938-3. [25] N. Saito and T. Suzuki, Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis, Applied Mathematics and Computation, 171 (2005), 72-90. doi: 10.1016/j.amc.2005.01.037. [26] D. L. Scharfetter and H. K. Gummel, Large signal analysis of a silicon Read diode, IEEE Transactions on Electron Devices, 16 (1969), 64-77. doi: 10.1109/T-ED.1969.16566.

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##### References:
 [1] N. J. Armstrong, K. J. Painter and J. A. Sherratt, A continuum approach to modelling cell-cell adhesion, Journal of Theoretical Biology, 243 (2006), 98-113. doi: 10.1016/j.jtbi.2006.05.030. [2] J. Barré, J. A. Carrillo, P. Degond, D. Peurichard and E. Zatorska, Particle interactions mediated by dynamical networks: Assessment of macroscopic descriptions, Journal of Nonlinear Science, 28 (2018), 235-268. doi: 10.1007/s00332-017-9408-z. [3] A. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710. doi: 10.1088/0951-7715/22/3/009. [4] M. Bessemoulin-Chatard and F. Filbet, A finite volume scheme for nonlinear degenerate parabolic equations, SIAM Journal on Scientific Computing, 34 (2012), 559-583. doi: 10.1137/110853807. [5] A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM Journal on Numerical Analysis, 46 (2008), 691-721. doi: 10.1137/070683337. [6] A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electronic Journal of Differential Equations, 2006 (2006), 1-32. [7] F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-balanced Schemes for Sources, Birkhäuser, 2004. doi: 10.1007/b93802. [8] J. A. Carrillo, A. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Communications in Computational Physics, 17 (2015), 233-258. doi: 10.4208/cicp.160214.010814a. [9] J. A. Carrillo, N. Kolbe and M. Lukacova-Medvidova, A hybrid mass-transport finite element method for Keller-Segel type systems, preprint, arXiv: 1709.07394, (2017). [10] J. A. Carrillo, Y. Huang and M. Schmidtchen, Zoology of a non-local cross-diffusion model for two species, SIAM Journal on Applied Mathematics, 78 (2018), 1078-1104. doi: 10.1137/17M1128782. [11] T. Cieślak and C. Morale-Rodrigo, Quasilinear non-uniformly parabolic-elliptic system modelling chemotaxis with volume filling effect. Existence and uniqueness of global-in-time solutions, Topological Methods in Nonlinear Analysis, 29 (2007), 361-381. [12] J. Dolbeault, P. Markowich and G. Jankowiak, Stationary solutions of Keller-Segel type crowd motion and herding models: Multiplicity and dynamical stability, Mathematics and Mechanics of Complex Systems, 3 (2015), 211-242. doi: 10.2140/memocs.2015.3.211. [13] R. Eymard, T. Gallouét and R. Herbin, Finite volume methods, Handbook of Numerical Analysis, 7 (2000), 713-1020. [14] T. Hillen and K. J. Painter, Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Applied Mathematics Quarterly, 10 (2002), 501-543. [15] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [16] T. Hillen, K. J. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete and Continuous Dynamical Systems Series B, 7 (2007), 125-144. doi: 10.3934/dcdsb.2007.7.125. [17] S. Jin and B. Yan, A class of asymptotic-preserving schemes for the Fokker-Planck-Landau equation, Journal of Computational Physics, 230 (2011), 6420-6437. doi: 10.1016/j.jcp.2011.04.002. [18] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [19] J. R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253. [20] J. Liu, L. Wang and Z. Zhou, Positivity-preserving and asymptotic preserving method for 2D Keller-Segel equations, Mathematics of Computation, 87 (2018), 1165-1189. doi: 10.1090/mcom/3250. [21] J. D. Murray, Mathematical Biology, vol. I: An Introduction, Springer, New York, 2002. [22] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Advances in Mathematical Sciences and Applications, 5 (1995), 581-601. [23] B. Perthame, Transport Equations in Biology, Birkhäuser Verlag, Basel, 2007. [24] A. B. Potapov and T. Hillen, Metastability in chemotaxis models, Journal of Dynamics and Differential Equations, 17 (2005), 293-330. doi: 10.1007/s10884-005-2938-3. [25] N. Saito and T. Suzuki, Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis, Applied Mathematics and Computation, 171 (2005), 72-90. doi: 10.1016/j.amc.2005.01.037. [26] D. L. Scharfetter and H. K. Gummel, Large signal analysis of a silicon Read diode, IEEE Transactions on Electron Devices, 16 (1969), 64-77. doi: 10.1109/T-ED.1969.16566.
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
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