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A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure

Published online by Cambridge University Press:  28 November 2014

José A. Carrillo
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Alina Chertock
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
Yanghong Huang*
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*Email A. Carrillo), Cherock), Huang)
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We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge.

Research Article
Copyright © Global Science Press Limited 2015 

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[1]Vázquez, J. L.. The Porous Medium Equation. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford, 2007. Mathematical Theory.Google Scholar
[2]Carrillo, J. A. and Toscani, G.. Asymptotic L 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J., 49(1):113142, 2000.CrossRefGoogle Scholar
[3]Otto, F.. The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations, 26(1-2):101174, 2001.CrossRefGoogle Scholar
[4]Keller, E. F. and Segel, L. A.. Initiation of slime mold aggregation viewed as an instability. J. Theoretical Biology, 26(3):399415, 1970.CrossRefGoogle ScholarPubMed
[5]Benedetto, D., Caglioti, E., and Pulvirenti, M.. A kinetic equation for granular media. RAIRO Modél. Math. Anal. Numér., 31(5):615641, 1997.CrossRefGoogle Scholar
[6]Carrillo, J. A., McCann, R. J., and Villani, C.. Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates. Rev. Mat. Iberoam., 19(3):9711018,2003.CrossRefGoogle Scholar
[7]Carrillo, J. A., McCann, R. J., and Villani, C.. Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal., 179(2):217263,2006.CrossRefGoogle Scholar
[8]Villani, C.. Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003.Google Scholar
[9]McCann, R. J.. A convexity principle for interacting gases. Adv. Math., 128(1):153179, 1997.CrossRefGoogle Scholar
[10]Ambrosio, L., Gigli, N., and Savaré, G.. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 2008.Google Scholar
[11]Benedetto, D., Caglioti, E., Carrillo, J. A., and Pulvirenti, M.. A non-Maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys., 91(5-6):979990, 1998.CrossRefGoogle Scholar
[12]Toscani, G.. One-dimensional kinetic models of granular flows. M2AN Math. Model. Numer. Anal., 34(6):12771291, 2000.CrossRefGoogle Scholar
[13]Li, H. and Toscani, G.. Long-time asymptotics of kinetic models of granular flows. Arch. Ration. Mech. Anal., 172(3):407428, 2004.CrossRefGoogle Scholar
[14]Topaz, C. M., Bertozzi, A. L., and Lewis, M. A.. A nonlocal continuum model for biological aggregation. Bull. Math. Biol., 68(7):16011623, 2006.CrossRefGoogle ScholarPubMed
[15]Bessemoulin-Chatard, M. and Filbet, F.. A finite volume scheme for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput., 34(5):B559B583, 2012.CrossRefGoogle Scholar
[16]Burger, M., Carrillo, J. A., and Wolfram, M.-T.. A mixed finite element method for nonlinear diffusion equations. Kinet. Relat. Models, 3(1):5983,2010.CrossRefGoogle Scholar
[17]Lie, K.-A. and Noelle, S.. On the artificial compression method for second-order nonoscil-latory central difference schemes for systems of conservation laws. SIAM J. Sci. Comput., 24(4):11571174,2003.CrossRefGoogle Scholar
[18]Nessyahu, H. and Tadmor, E.. Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys., 87(2):408463,1990.CrossRefGoogle Scholar
[19]Sweby, P. K.. High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal., 21(5):9951011, 1984.CrossRefGoogle Scholar
[20]van Leer, B.. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method. J. Comput. Phys., 32(1):101136,1979.CrossRefGoogle Scholar
[21]Gottlieb, S., Shu, C.-W., and Tadmor, E.. Strong stability-preserving high-order time discretization methods. SIAM Rev., 43:89112, 2001.CrossRefGoogle Scholar
[22]von zur Gathen, J. and Gerhard, J.. Modern Computer Algebra. Cambridge University Press, Cambridge, second edition, 2003.Google Scholar
[23]Saff, E. B. and Totik, V.. Logarithmic Potentials with External Fields, volume 316 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences). Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom.Google Scholar
[24]Carrillo, J. A., Ferreira, L. C. F., and Precioso, J. C.. A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity. Adv. Math., 231(1):306327, 2012.CrossRefGoogle Scholar
[25]Burger, M., Fetecau, R., and Huang, Y. Stationary states and asymptotic behaviour of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13(1):397424, 2014.CrossRefGoogle Scholar
[26]Burger, M., di Francesco, M., and Franek, M.. Stationary states of quadratic diffusion equations with long-range attraction. Commun. Math. Sci., 11(3):709738,2013.CrossRefGoogle Scholar
[27]Balagué, D., Carrillo, J. A., Laurent, T., and Raoul, G.. Dimensionality of local minimizers of the interaction energy. Arch. Ration. Mech. Anal., 209(3):10551088,2013.CrossRefGoogle Scholar
[28]Fellner, K. and Raoul, G.. Stability of stationary states of non-local equations with singular interaction potentials. Math. Comput. Modelling, 53(7-8):14361450,2011.CrossRefGoogle Scholar
[29]Velázquez, J. J. L.. Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions. SIAM J. Appl. Math., 64(4):11981223,2004.CrossRefGoogle Scholar
[30]Calvez, V. and Carrillo, J. A.. Volume effects in the Keller-Segel model: Energy estimates preventing blow-up. J. Math. Pures Appl. (9), 86(2):155175,2006.CrossRefGoogle Scholar
[31]Blanchet, A., Carrillo, J. A., and Masmoudi, N.. Infinite time aggregation for the critical Patlak-Keller-Segel model in ℝ2. Comm. Pure Appl. Math., 61(10):14491481,2008.CrossRefGoogle Scholar
[32]Blanchet, A., Carlen, E. A., and Carrillo, J. A.. Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model. J. Funct. Anal., 262(5):21422230,2012.CrossRefGoogle Scholar
[33]Blanchet, A., Carrillo, J. A., and Laurençot, P.. Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions. Calc. Var. Partial Differential Equations, 35(2):133168,2009.CrossRefGoogle Scholar
[34]Yao, Y. and Bertozzi, A. L.. Blow-up dynamics for the aggregation equation with degenerate diffusion. Physica D: Nonlinear Phenomena, 260(0):7789,2013.CrossRefGoogle Scholar
[35]Campos, J.F. and Dolbeault, J.. Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane. Comm. Partial Differential Equations, 39(5):806841,2014.CrossRefGoogle Scholar
[36]Ströhmer, G.. Stationary states and moving planes. In Parabolic and Navier-Stokes equations. Part 2, volume 81 of Banach Center Publ., pages 501513. Polish Acad. Sci. Inst. Math., Warsaw, 2008.Google Scholar
[37]Levine, H., Rappel, W.-J., and Cohen, I.. Self-organization in systems of self-propelled particles. Phys. Rev. E, 63:017101, Dec 2000.CrossRefGoogle ScholarPubMed
[38]D'Orsogna, M. R., Chuang, Y.-L., Bertozzi, A. L., and Chayes, L. S.. Self-propelled particles with soft-core interactions: Patterns, stability, and collapse. Phys. Rev. Lett., 96(10):104302, 2006.Google Scholar
[39]Fellner, K. and Raoul, G.. Stable stationary states of non-local interaction equations. Math. Models Methods Appl. Sci., 20(12):22672291, 2010.CrossRefGoogle Scholar
[40]Fetecau, R. C., Huang, Y., and Kolokolnikov, T.. Swarm dynamics and equilibria for a nonlocal aggregation model. Nonlinearity, 24(10):26812716,2011.CrossRefGoogle Scholar
[41]Carrillo, J. A., D'Orsogna, M. R., and Panferov, V.. Double milling in self-propelled swarms from kinetic theory. Kinet. Relat. Models, 2(2):363378,2009.CrossRefGoogle Scholar
[42]Carrillo, J. A., Martin, S., and Panferov, V.. A new interaction potential for swarming models. Physica D: Nonlinear Phenomena, 260(0):112126,2013.CrossRefGoogle Scholar
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