A blob method for the aggregation equation
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- by Katy Craig and Andrea L. Bertozzi;
- Math. Comp. 85 (2016), 1681-1717
- DOI: https://doi.org/10.1090/mcom3033
- Published electronically: December 4, 2015
Abstract:
Motivated by classical vortex blob methods for the Euler equations, we develop a numerical blob method for the aggregation equation. This provides a counterpoint to existing literature on particle methods. By regularizing the velocity field with a mollifier or “blob function”, the blob method has a faster rate of convergence and allows a wider range of admissible kernels. In fact, we prove arbitrarily high polynomial rates of convergence to classical solutions, depending on the choice of mollifier. The blob method conserves mass and the corresponding particle system is energy decreasing for a regularized free energy functional and preserves the Wasserstein gradient flow structure. We consider numerical examples that validate our predicted rate of convergence and illustrate qualitative properties of the method.References
- Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient flows in metric spaces and in the space of probability measures, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. MR 2401600
- Luigi Ambrosio and Sylvia Serfaty, A gradient flow approach to an evolution problem arising in superconductivity, Comm. Pure Appl. Math. 61 (2008), no. 11, 1495–1539. MR 2444374, DOI 10.1002/cpa.20223
- Christopher Anderson and Claude Greengard, On vortex methods, SIAM J. Numer. Anal. 22 (1985), no. 3, 413–440. MR 787568, DOI 10.1137/0722025
- D. Balagué, J. A. Carrillo, T. Laurent, and G. Raoul, Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal. 209 (2013), no. 3, 1055–1088. MR 3067832, DOI 10.1007/s00205-013-0644-6
- D. Balagué, J. A. Carrillo, T. Laurent, and G. Raoul, Nonlocal interactions by repulsive-attractive potentials: radial ins/stability, Phys. D 260 (2013), 5–25. MR 3143991, DOI 10.1016/j.physd.2012.10.002
- Daniel Balagué, José A. Carrillo, and Yao Yao, Confinement for repulsive-attractive kernels, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 5, 1227–1248. MR 3199778, DOI 10.3934/dcdsb.2014.19.1227
- J. Thomas Beale, A convergent $3$-D vortex method with grid-free stretching, Math. Comp. 46 (1986), no. 174, 401–424, S15–S20. MR 829616, DOI 10.1090/S0025-5718-1986-0829616-6
- J. Thomas Beale and Andrew Majda, Vortex methods. I. Convergence in three dimensions, Math. Comp. 39 (1982), no. 159, 1–27. MR 658212, DOI 10.1090/S0025-5718-1982-0658212-5
- J. Thomas Beale and Andrew Majda, Vortex methods. II. Higher order accuracy in two and three dimensions, Math. Comp. 39 (1982), no. 159, 29–52. MR 658213, DOI 10.1090/S0025-5718-1982-0658213-7
- Andrea L. Bertozzi and Jeremy Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Commun. Math. Sci. 8 (2010), no. 1, 45–65. MR 2655900
- Andrea L. Bertozzi, José A. Carrillo, and Thomas Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity 22 (2009), no. 3, 683–710. MR 2480108, DOI 10.1088/0951-7715/22/3/009
- Andrea L. Bertozzi, John B. Garnett, and Thomas Laurent, Characterization of radially symmetric finite time blowup in multidimensional aggregation equations, SIAM J. Math. Anal. 44 (2012), no. 2, 651–681. MR 2914245, DOI 10.1137/11081986X
- Andrea L. Bertozzi, Thomas Laurent, and Flavien Léger, Aggregation and spreading via the Newtonian potential: the dynamics of patch solutions, Math. Models Methods Appl. Sci. 22 (2012), no. suppl. 1, 1140005, 39. MR 2974185, DOI 10.1142/S0218202511400057
- Andrea L. Bertozzi, Theodore Kolokolnikov, Hui Sun, David Uminsky, and James von Brecht, Ring patterns and their bifurcations in a nonlocal model of biological swarms, Commun. Math. Sci. 13 (2015), no. 4, 955–985. MR 3325085, DOI 10.4310/CMS.2015.v13.n4.a6
- H. S. Bhat and R. C. Fetecau, A Hamiltonian regularization of the Burgers equation, J. Nonlinear Sci. 16 (2006), no. 6, 615–638. MR 2271428, DOI 10.1007/s00332-005-0712-7
- H. S. Bhat and R. C. Fetecau, Stability of fronts for a regularization of the Burgers equation, Quart. Appl. Math. 66 (2008), no. 3, 473–496. MR 2445524, DOI 10.1090/S0033-569X-08-01099-X
- H. S. Bhat and R. C. Fetecau, The Riemann problem for the Leray-Burgers equation, J. Differential Equations 246 (2009), no. 10, 3957–3979. MR 2514732, DOI 10.1016/j.jde.2009.01.006
- Giovanni A. Bonaschi, José A. Carrillo, Marco Di Francesco, and Mark A. Peletier, Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D, ESAIM Control Optim. Calc. Var. 21 (2015), no. 2, 414–441. MR 3348406, DOI 10.1051/cocv/2014032
- Peter N. Brown, Alan C. Hindmarsh, and George D. Byrne, DVODE: Variable-coefficient ordinary differential equation solver, 1975.
- J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent, and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J. 156 (2011), no. 2, 229–271. MR 2769217, DOI 10.1215/00127094-2010-211
- J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, and D. Slepčev, Confinement in nonlocal interaction equations, Nonlinear Anal. 75 (2012), no. 2, 550–558. MR 2847439, DOI 10.1016/j.na.2011.08.057
- José A. Carrillo, Alina Chertock, and Yanghong Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys. 17 (2015), no. 1, 233–258. MR 3372289, DOI 10.4208/cicp.160214.010814a
- José Antonio Carrillo, Michel Chipot, and Yanghong Huang, On global minimizers of repulsive-attractive power-law interaction energies, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), no. 2028, 20130399, 13. MR 3268056, DOI 10.1098/rsta.2013.0399
- J. A. Carrillo, Y.-P. Choi, and M. Hauray, The derivation of swarming models: mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation Series, CISM International Centre for Mechanical Sciences 553 (2014), 1–46.
- Rustum Choksi, Razvan C. Fetecau, and Ihsan Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Ann. Inst. H. Poincaré C Anal. Non Linéaire 32 (2015), no. 6, 1283–1305. MR 3425263, DOI 10.1016/j.anihpc.2014.09.004
- Alexandre Joel Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785–796. MR 395483, DOI 10.1017/S0022112073002016
- Philippe Clément and Wolfgang Desch, A Crandall-Liggett approach to gradient flows in metric spaces, J. Abstr. Differ. Equ. Appl. 1 (2010), no. 1, 46–60. MR 2747655
- G.-H. Cottet and P.-A. Raviart, Particle methods for the one-dimensional Vlasov-Poisson equations, SIAM J. Numer. Anal. 21 (1984), no. 1, 52–76. MR 731212, DOI 10.1137/0721003
- Hongjie Dong, The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions, Comm. Math. Phys. 304 (2011), no. 3, 649–664. MR 2794542, DOI 10.1007/s00220-011-1237-6
- J. P. K. Doye, D. J. Wales, and R. S. Berry, The effect of the range of the potential on the structures of clusters, J. Chem. Phys. 103 (1995), 4234–4249.
- Qiang Du and Ping Zhang, Existence of weak solutions to some vortex density models, SIAM J. Math. Anal. 34 (2003), no. 6, 1279–1299. MR 2000970, DOI 10.1137/S0036141002408009
- Yong Duan and Jian-Guo Liu, Convergence analysis of the vortex blob method for the $b$-equation, Discrete Contin. Dyn. Syst. 34 (2014), no. 5, 1995–2011. MR 3124723, DOI 10.3934/dcds.2014.34.1995
- Jeff D. Eldredge, Tim Colonius, and Anthony Leonard, A vortex particle method for two-dimensional compressible flow, J. Comput. Phys. 179 (2002), no. 2, 371–399. MR 1911371, DOI 10.1006/jcph.2002.7060
- Klemens Fellner and Gaël Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci. 20 (2010), no. 12, 2267–2291. MR 2755500, DOI 10.1142/S0218202510004921
- Klemens Fellner and Gaël Raoul, Stability of stationary states of non-local equations with singular interaction potentials, Math. Comput. Modelling 53 (2011), no. 7-8, 1436–1450. MR 2782822, DOI 10.1016/j.mcm.2010.03.021
- R. C. Fetecau and Y. Huang, Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Phys. D 260 (2013), 49–64. MR 3143993, DOI 10.1016/j.physd.2012.11.004
- R. C. Fetecau, Y. Huang, and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity 24 (2011), no. 10, 2681–2716. MR 2834242, DOI 10.1088/0951-7715/24/10/002
- Gerald B. Folland, Real analysis, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. Modern techniques and their applications; A Wiley-Interscience Publication. MR 1681462
- Jonathan Goodman, Convergence of the random vortex method, Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986) IMA Vol. Math. Appl., vol. 9, Springer, New York, 1987, pp. 99–106. MR 914987, DOI 10.1007/978-1-4684-6347-7_{7}
- Jonathan Goodman, Thomas Y. Hou, and John Lowengrub, Convergence of the point vortex method for the $2$-D Euler equations, Comm. Pure Appl. Math. 43 (1990), no. 3, 415–430. MR 1040146, DOI 10.1002/cpa.3160430305
- Ole Hald and Vincenza Mauceri del Prete, Convergence of vortex methods for Euler’s equations, Math. Comp. 32 (1978), no. 143, 791–809. MR 492039, DOI 10.1090/S0025-5718-1978-0492039-1
- Ole H. Hald, Convergence of vortex methods for Euler’s equations. II, SIAM J. Numer. Anal. 16 (1979), no. 5, 726–755. MR 543965, DOI 10.1137/0716055
- Thomas Y. Hou and John Lowengrub, Convergence of the point vortex method for the $3$-D Euler equations, Comm. Pure Appl. Math. 43 (1990), no. 8, 965–981. MR 1075074, DOI 10.1002/cpa.3160430803
- Y. Huang, T. P. Witelski, and A. L. Bertozzi, Anomalous exponents of self-similar blow-up solutions to an aggregation equation in odd dimensions, Appl. Math. Lett. 25 (2012), no. 12, 2317–2321. MR 2967836, DOI 10.1016/j.aml.2012.06.023
- Yanghong Huang and Andrea Bertozzi, Asymptotics of blowup solutions for the aggregation equation, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 4, 1309–1331. MR 2899948, DOI 10.3934/dcdsb.2012.17.1309
- Yanghong Huang and Andrea L. Bertozzi, Self-similar blowup solutions to an aggregation equation in $\mathbf R^n$, SIAM J. Appl. Math. 70 (2010), no. 7, 2582–2603. MR 2678052, DOI 10.1137/090774495
- John D. Hunter, Matplotlib: a 2D graphics environment, Computing in Science and Engineering 9 (2007), 90–95.
- Francois James and Nicolas Vauchelet, Numerical methods for one-dimensional aggregation equations, SIAM J. Numer. Anal. 53 (2015), no. 2, 895–916. MR 3327358, DOI 10.1137/140959997
- E. Jones, T. Oliphant, Pearu Peterson, et al., SciPy: Open source scientific tools for Python, 2001–.
- T. Kolokolnikov, H. Sun, D. Uminsky, and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E 84 (2011), no. 1.
- Fanghua Lin and Ping Zhang, On the hydrodynamic limit of Ginzburg-Landau vortices, Discrete Contin. Dynam. Systems 6 (2000), no. 1, 121–142. MR 1739596, DOI 10.3934/dcds.2000.6.121
- Ding-Gwo Long, Convergence of the random vortex method in two dimensions, J. Amer. Math. Soc. 1 (1988), no. 4, 779–804. MR 958446, DOI 10.1090/S0894-0347-1988-0958446-1
- Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
- C. Marchioro and M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys. 84 (1982), no. 4, 483–503. MR 667756
- Nader Masmoudi and Ping Zhang, Global solutions to vortex density equations arising from sup-conductivity, Ann. Inst. H. Poincaré C Anal. Non Linéaire 22 (2005), no. 4, 441–458 (English, with English and French summaries). MR 2145721, DOI 10.1016/j.anihpc.2004.07.002
- A. Mogilner, L. Edelstein-Keshet, L. Bent, and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation, J. Math. Biol. 47 (2003), no. 4, 353–389. MR 2024502, DOI 10.1007/s00285-003-0209-7
- Alexander Mogilner and Leah Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol. 38 (1999), no. 6, 534–570. MR 1698215, DOI 10.1007/s002850050158
- Greg Norgard and Kamran Mohseni, A regularization of the Burgers equation using a filtered convective velocity, J. Phys. A 41 (2008), no. 34, 344016, 21. MR 2456353, DOI 10.1088/1751-8113/41/34/344016
- Greg Norgard and Kamran Mohseni, On the convergence of the convectively filtered Burgers equation to the entropy solution of the inviscid Burgers equation, Multiscale Model. Simul. 7 (2009), no. 4, 1811–1837. MR 2539200, DOI 10.1137/080735485
- L. Perea, G. Gómez, and P. Elosegui, Extension of the Cucker–Smale control law to space flight formations, AIAA J. of Guidance, Control, and Dynamics 32 (2009), 527–537.
- Frédéric Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Methods Appl. Anal. 9 (2002), no. 4, 533–561. MR 2006604, DOI 10.4310/MAA.2002.v9.n4.a4
- Gaël Raoul, Nonlocal interaction equations: stationary states and stability analysis, Differential Integral Equations 25 (2012), no. 5-6, 417–440. MR 2951735
- M. C. Rechtsman, F. H. Stillinger, and S. Torquato, Optimized interactions for targeted self- assembly: application to a honeycomb lattice, Phys. Rev. Lett. 95 (2005), no. 22.
- Louis F. Rossi, Resurrecting core spreading vortex methods: a new scheme that is both deterministic and convergent, SIAM J. Sci. Comput. 17 (1996), no. 2, 370–397. MR 1374286, DOI 10.1137/S1064827593254397
- Hui Sun, David Uminsky, and Andrea L. Bertozzi, A generalized Birkhoff-Rott equation for two-dimensional active scalar problems, SIAM J. Appl. Math. 72 (2012), no. 1, 382–404. MR 2888349, DOI 10.1137/110819883
- Hui Sun, David Uminsky, and Andrea L. Bertozzi, Stability and clustering of self-similar solutions of aggregation equations, J. Math. Phys. 53 (2012), no. 11, 115610, 18. MR 3026555, DOI 10.1063/1.4745180
- Chad M. Topaz and Andrea L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), no. 1, 152–174. MR 2111591, DOI 10.1137/S0036139903437424
- Chad M. Topaz, Andrea L. Bertozzi, and Mark A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol. 68 (2006), no. 7, 1601–1623. MR 2257718, DOI 10.1007/s11538-006-9088-6
- S. van der Walt, S. C. Colbert, and G. Varoquaux, The NumPy array: A structure for efficient numerical computation, Computing in Science and Engineering 13 (2011), 23–30.
- D. J. Wales, Energy landscapes of clusters bound by short-ranged potentials, Chem. Eur. J. Chem. Phys. 11 (2010), 2491–2494.
- Weinan E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Phys. D 77 (1994), no. 4, 383–404. MR 1297726, DOI 10.1016/0167-2789(94)90298-4
- Yao Yao and Andrea L. Bertozzi, Blow-up dynamics for the aggregation equation with degenerate diffusion, Phys. D 260 (2013), 77–89. MR 3143995, DOI 10.1016/j.physd.2013.01.009
- Lung-an Ying and Pingwen Zhang, Vortex methods, Mathematics and its Applications, vol. 381, Kluwer Academic Publishers, Dordrecht; Science Press Beijing, Beijing, 1997. MR 1705273
Bibliographic Information
- Katy Craig
- Affiliation: Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, California 90095-1555
- Email: kcraig@math.ucla.edu
- Andrea L. Bertozzi
- Affiliation: Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, California 90095-1555
- MR Author ID: 265966
- Email: bertozzi@math.ucla.edu
- Received by editor(s): May 29, 2014
- Received by editor(s) in revised form: December 13, 2014, and January 7, 2015
- Published electronically: December 4, 2015
- Additional Notes: This work was supported by NSF grants CMMI-1435709, DMS-0907931, DMS-1401867, and EFRI-1024765, as well as NSF grant 0932078 000, which supported Craig’s visit and Bertozzi’s residence at the Mathematical Sciences Research Institute during Fall 2013.
- © Copyright 2015 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
- Journal: Math. Comp. 85 (2016), 1681-1717
- MSC (2010): Primary 35Q35, 35Q82, 65M15, 82C22
- DOI: https://doi.org/10.1090/mcom3033
- MathSciNet review: 3471104