Numerical study of a particle method for gradient flows

José Antonio Carrillo; Yanghong Huang; Francesco Saverio Patacchini; Gershon Wolansky

doi:10.3934/krm.2017025

Article Contents

2017, Volume 10, Issue 3: 613-641. Doi: 10.3934/krm.2017025 open access

This issue Previous Article Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models Next Article Asymptotic preserving and time diminishing schemes for rarefied gas dynamic

Numerical study of a particle method for gradient flows

1.
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
2.
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
3.
Mathematics Department, Technion--Israel Institute of Technology, Haifa 32000, Israel

* Corresponding author: J. A. Carrillo
* Corresponding author: J. A. Carrillo

Received: May 31, 2016

Revised: September 30, 2016

Published: September 2017

The first, second and third authors are supported by Engineering and Physical Sciences Research Council grant EP/K008404/1. The first author is also supported by the Royal Society through a Wolfson Research Merit Award. The last author is supported by ISF grant 998/5

Abstract Full Text(HTML) Figure(11) Related Papers Cited by

Abstract

We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.

Keywords:

Mathematics Subject Classification: Primary: 65M12; Secondary: 35K05.

Citation:

FullText(HTML)

Figure 1. The heat equation

Download: Full-size image PowerPoint slide

Figure 2. The porous medium equation with $m=2$

Download: Full-size image PowerPoint slide

Figure 3. The linear Fokker-Planck equation with $\Delta t = 10^{-5}$ --Stabilisation in time of the scheme (rate of convergence to the discrete steady state)

Download: Full-size image PowerPoint slide

Figure 4. The nonlinear Fokker-Planck equation with $m=2$ and $\Delta t = \frac{0.1}{N^2}$ -Stabilisation in time of the scheme (rate of convergence to the discrete steady state)

Download: Full-size image PowerPoint slide

Figure 5. The particles' positions for the two-dimensional heat equation for $N=100$ with $\Delta t = 10^{-4}$

Download: Full-size image PowerPoint slide

Figure 6. Accuracy for the two-dimensional heat equation with $\Delta t = 10^{-4}$

Download: Full-size image PowerPoint slide

Figure 7. The modified Keller-Segel equation with $\chi = 1.5$ for $N=50$

Download: Full-size image PowerPoint slide

Figure 8. The blow-up of the modified Keller-Segel equation with $\chi = 1.5$

Download: Full-size image PowerPoint slide

Figure 9. Blow-up formation for the modified Keller-Segel equation with two initial Gaussian bumps

Download: Full-size image PowerPoint slide

Figure 10. The modified nonlinear Keller-Segel equation with $\chi = 1.4$ for different choices of $m$, for $N=50$ at $T=4$ with $\Delta t = 10^{-5}$

Download: Full-size image PowerPoint slide

Figure 11. Compactly supported potential $W(x) = -c\max(1-|x|,0) + c$ with nonlinear diffusion with $c = 8$ and $m = 3$, for $N=80$ with $\Delta t = 10^{-5}$

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

	L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser Basel, 2005. doi: 10.1007/978-3-7643-8722-8.
	L. Ambrosio and G. Savaré, Gradient flows of probability measures, In Handbook of Differential Equations: Evolutionary Equations. North-Holland, 3 (2007), 1-136. doi: 10.1016/S1874-5717(07)80004-1.
	H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics. Pitman Advanced Publishing Program, 1984.
	J. -P. Aubin and A. Cellina, Differential Inclusions, Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 1984. doi: 10.1007/978-3-642-69512-4.
	J.-D. Benamou , G. Carlier , Q. Mérigot and E. Oudet , Discretization of functionals involving the Monge-Ampère operator, Numer. Math., 134 (2016) , 611-636. doi: 10.1007/s00211-015-0781-y.
	D. Benedetto , E. Caglioti , J. A. Carrillo and M. Pulvirenti , A non-Maxwellian steady distribution for one-dimensional granular media, J. Statist. Phys., 91 (1998) , 979-990. doi: 10.1023/A:1023032000560.
	M. Bessemoulin-Chatard and F. Filbet , A finite volume scheme for nonlinear degenerate parabolic equations, SIAM J. Sci. Comput., 34 (2012) , B559-B583. doi: 10.1137/110853807.
	A. Blanchet , V. Calvez and J. A. Carrillo , Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008) , 691-721. doi: 10.1137/070683337.
	H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies. Elsevier Science, 1973.
	V. Calvez and T. Gallouët , Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up, Discrete Contin. Dyn. Syst., 36 (2016) , 1175-1208. doi: 10.3934/dcds.2016.36.1175.
	V. Calvez , B. Perthame and M. Sharifi-tabar , Modified Keller-Segel system and critical mass for the log interaction kernel, Contemp. Math., 429 (2007) , 45-62. doi: 10.1090/conm/429/08229.
	J. A. Carrillo , A. Chertock and Y. Huang , A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015) , 233-258. doi: 10.4208/cicp.160214.010814a.
	J. A. Carrillo, Y. -P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, In Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation. Springer Vienna, 553 (2014), 1-46. doi: 10.1007/978-3-7091-1785-9_1.
	J. A. Carrillo , M. Di Francesco and G. Toscani , Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering, Proc. Amer. Math. Soc., 135 (2007) , 353-363. doi: 10.1090/S0002-9939-06-08594-7.
	J. A. Carrillo , R. J. McCann and C. Villani , Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19 (2003) , 971-1018. doi: 10.4171/RMI/376.
	J. A. Carrillo and J. S. Moll , Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM J. Sci. Comput., 31 (2009) , 4305-4329. doi: 10.1137/080739574.
	J. A. Carrillo , F. S. Patacchini , P. Sternberg and G. Wolansky , Convergence of a particle method for diffusive gradient flows in one dimension, SIAM J. Math. Anal., 48 (2016) , 3708-3741. doi: 10.1137/16M1077210.
	P. Degond and F. J. Mustieles , A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Statist. Comput., 11 (1990) , 293-310. doi: 10.1137/0911018.
	L. Gosse and G. Toscani , Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput., 28 (2006) , 1203-1227. doi: 10.1137/050628015.
	S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2000. doi: 10.1007/BFb0103945.
	R. Jordan , D. Kinderlehrer and F. Otto , The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998) , 1-17. doi: 10.1137/S0036141096303359.
	O. Junge, H. Osberger and D. Matthes, A fully discrete variational scheme for solving nonlinear Fokker-Planck equations in higher space dimensions, Preprint, arXiv: 1509.07721.
	P.-L. Lions and S. Mas-Gallic , Une méthode particulaire déterministe pour des équations diffusives non linéaires, C. R. Acad. Sci. Paris Sér. Ⅰ Math., 332 (2001) , 369-376. doi: 10.1016/S0764-4442(00)01795-X.
	S. Mas-Gallic , The diffusion velocity method: A deterministic way of moving the nodes for solving diffusion equations, Transp. Theory Stat. Phys., 31 (2002) , 595-605. doi: 10.1081/TT-120015516.
	R. J. McCann , A convexity principle for interacting gases, Adv. Math., 128 (1997) , 153-179. doi: 10.1006/aima.1997.1634.
	R. J. McCann, A Convexity Theory for Interacting Gases and Equilibrium Crystals, PhD thesis, Princeton University, 1994.
	A. Mielke, On evolutionary Gamma-convergence for gradient systems, In Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. Springer, 3 (2016), 187-249. doi: 10.1007/978-3-319-26883-5_3.
	H. Osberger and D. Matthes , A convergent Lagrangian discretization for a nonlinear fourth order equation, Found. Comput. Math., (2015) , 1-54. doi: 10.1007/s10208-015-9284-6.
	H. Osbergers and D. Matthes , Convergence of a fully discrete variational scheme for a thin-film equation, Accepted in Radon Ser. Comput. Appl. Math., (2015) .
	H. Osberger and D. Matthes , Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM Math. Model. Numer. Anal., 48 (2014) , 697-726. doi: 10.1051/m2an/2013126.
	F. Otto , The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001) , 101-174. doi: 10.1081/PDE-100002243.
	G. Russo , A particle method for collisional kinetic equations. Ⅰ. Basic theory and one-dimensional results, J. Comput. Phys., 87 (1990) , 270-300. doi: 10.1016/0021-9991(90)90254-X.
	G. Russo , Deterministic diffusion of particles, Comm. Pure Appl. Math., 43 (1990) , 697-733. doi: 10.1002/cpa.3160430602.
	E. Sandier and S. Serfaty , Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004) , 1627-1672. doi: 10.1002/cpa.20046.
	S. Serfaty , Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst, 31 (2011) , 1427-1451. doi: 10.3934/dcds.2011.31.1427.
	C. M. Topaz , A. L. Bertozzi and M. A. Lewis , A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006) , 1601-1623. doi: 10.1007/s11538-006-9088-6.
	J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.
	C. Villani, Topics in Optimal Transportation, Graduate studies in mathematics. American Mathematical Society, Providence (R. I. ), 2003. doi: 10.1090/gsm/058.