Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation

Previous Article
Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line
DCDS-B Home
This Issue
Next Article
An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip

November 2015, 20(9): 3013-3027. doi: 10.3934/dcdsb.2015.20.3013

Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation

1.	1-Université de Toulouse; UPS, INSA, UT1, UTM, 2-CNRS, Institut de Mathématiques de Toulouse, F-31062 Toulouse, France

Received July 2014 Revised May 2015 Published September 2015

Abstract
Related Papers

This paper is devoted to study the existence of solutions of hydrodynamic model for systems of self-propelled particles subject to alignment and volume exclusion interactions. On one hand, we prove the existence of solutions by using the modified Garlerkin method for quasi-linear parabolic simulations. On the other hand, we also perform simulations to compare theoretical and numerical results. The numerical results show that the numerical solutions exist for short time in some cases of coefficients.

Keywords: vortex problem., Non-conservative equations, existence of solution, numerical solution.

Mathematics Subject Classification: 35Q80, 35L60, 82C22, 82C70, 92D5.

Citation: Thi-Bich-Ngoc Mac. Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3013-3027. doi: 10.3934/dcdsb.2015.20.3013

References:

[1]	A. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents,, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1249. doi: 10.3934/dcdsb.2014.19.1249. Google Scholar
[2]	A. Baskaran and M. C. Marchetti, Hydrodynamics of self-propelled hard rods,, Phys. Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.011920. Google Scholar
[3]	E. Bertin, M. Droz and G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis,, J. Phys. A: Math. Theor., 42 (2009). doi: 10.1088/1751-8113/42/44/445001. Google Scholar
[4]	A. Cziròk and T. Vicsek, Collective behavior of interacting self-propelled particles,, Physica A, 281 (2000), 17. doi: 10.1016/S0378-4371(00)00013-3. Google Scholar
[5]	P. Degond, G. Dimarco , T. B. N. Mac and N. Wang, Macroscopic models of collective motion with repulsion,, Communications in Mathematical Sciences, 13 (2015), 1615. doi: 10.4310/CMS.2015.v13.n6.a12. Google Scholar
[6]	P. Degond, A. Frouvelle and J.-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles,, J. Nonlinear Sci., 23 (2013), 427. doi: 10.1007/s00332-012-9157-y. Google Scholar
[7]	P. Degond, A. Frouvelle, J.-G. Liu, S. Motsch and L. Navoret, Macroscopic models of collective motion and self-organization,, Séminaire Laurent Schwartz — EDP et applications, (): 2012. doi: 10.5802/slsedp.32. Google Scholar
[8]	P. Degond, A. Frouvelle and J.-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics,, Archive for Rational Mechanics and Analysis, 216 (2015), 63. doi: 10.1007/s00205-014-0800-7. Google Scholar
[9]	P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction,, Math. Models Methods Appl. Sci., 18 (2008), 1193. doi: 10.1142/S0218202508003005. Google Scholar
[10]	P. Degond, J.-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory,, Methods Appl. Anal., 20 (2013), 89. doi: 10.4310/MAA.2013.v20.n2.a1. Google Scholar
[11]	A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters,, Math. Mod. Meth. Appl. Sci., 22 (2012). doi: 10.1142/S021820251250011X. Google Scholar
[12]	A. Frouvelle and J.-G. Liu, Dynamics in a kinetic model of oriented particles with phase transition,, SIAM J. Math. Anal., 44 (2012), 791. doi: 10.1137/110823912. Google Scholar
[13]	S. Henkes, Y. Fily and M. C. Marchetti, Active jamming: Self-propelled soft particles at high density,, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.040301. Google Scholar
[14]	S. Motsch and L. Navoret, Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior,, Multiscale Model. Simul., 9 (2011), 1253. doi: 10.1137/100794067. Google Scholar
[15]	F. Peruani, A. Deutsch and M. Bär, Nonequilibrium clustering of self-propelled rods,, Phys. Rev. E, 74 (2006). doi: 10.1103/PhysRevE.74.030904. Google Scholar
[16]	V. I. Ratushnaya, D. Bedeaux, V. L. Kulinskii and A. V. Zvelindovsky, Collective behavior of self propelling particles with kinematic constraints: the relations between the discrete and the continuous description,, Phys. A, 381 (2007), 39. doi: 10.1016/j.physa.2007.03.045. Google Scholar
[17]	M. E. Taylor, Partial Differential Equations III,, Second edition. Applied Mathematical Sciences, (2011). doi: 10.1007/978-1-4419-7049-7. Google Scholar
[18]	Y. Tu, J. Toner and M. Ulm, Sound waves and the absence of Galilean invariance in flocks,, Phys. Rev. Lett., 80 (1998), 4819. doi: 10.1103/PhysRevLett.80.4819. Google Scholar
[19]	T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles,, Phys. Rev. Lett., 75 (1995), 1226. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

show all references

References:

[1]	A. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents,, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1249. doi: 10.3934/dcdsb.2014.19.1249. Google Scholar
[2]	A. Baskaran and M. C. Marchetti, Hydrodynamics of self-propelled hard rods,, Phys. Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.011920. Google Scholar
[3]	E. Bertin, M. Droz and G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis,, J. Phys. A: Math. Theor., 42 (2009). doi: 10.1088/1751-8113/42/44/445001. Google Scholar
[4]	A. Cziròk and T. Vicsek, Collective behavior of interacting self-propelled particles,, Physica A, 281 (2000), 17. doi: 10.1016/S0378-4371(00)00013-3. Google Scholar
[5]	P. Degond, G. Dimarco , T. B. N. Mac and N. Wang, Macroscopic models of collective motion with repulsion,, Communications in Mathematical Sciences, 13 (2015), 1615. doi: 10.4310/CMS.2015.v13.n6.a12. Google Scholar
[6]	P. Degond, A. Frouvelle and J.-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles,, J. Nonlinear Sci., 23 (2013), 427. doi: 10.1007/s00332-012-9157-y. Google Scholar
[7]	P. Degond, A. Frouvelle, J.-G. Liu, S. Motsch and L. Navoret, Macroscopic models of collective motion and self-organization,, Séminaire Laurent Schwartz — EDP et applications, (): 2012. doi: 10.5802/slsedp.32. Google Scholar
[8]	P. Degond, A. Frouvelle and J.-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics,, Archive for Rational Mechanics and Analysis, 216 (2015), 63. doi: 10.1007/s00205-014-0800-7. Google Scholar
[9]	P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction,, Math. Models Methods Appl. Sci., 18 (2008), 1193. doi: 10.1142/S0218202508003005. Google Scholar
[10]	P. Degond, J.-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory,, Methods Appl. Anal., 20 (2013), 89. doi: 10.4310/MAA.2013.v20.n2.a1. Google Scholar
[11]	A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters,, Math. Mod. Meth. Appl. Sci., 22 (2012). doi: 10.1142/S021820251250011X. Google Scholar
[12]	A. Frouvelle and J.-G. Liu, Dynamics in a kinetic model of oriented particles with phase transition,, SIAM J. Math. Anal., 44 (2012), 791. doi: 10.1137/110823912. Google Scholar
[13]	S. Henkes, Y. Fily and M. C. Marchetti, Active jamming: Self-propelled soft particles at high density,, Phys. Rev. E, 84 (2011). doi: 10.1103/PhysRevE.84.040301. Google Scholar
[14]	S. Motsch and L. Navoret, Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior,, Multiscale Model. Simul., 9 (2011), 1253. doi: 10.1137/100794067. Google Scholar
[15]	F. Peruani, A. Deutsch and M. Bär, Nonequilibrium clustering of self-propelled rods,, Phys. Rev. E, 74 (2006). doi: 10.1103/PhysRevE.74.030904. Google Scholar
[16]	V. I. Ratushnaya, D. Bedeaux, V. L. Kulinskii and A. V. Zvelindovsky, Collective behavior of self propelling particles with kinematic constraints: the relations between the discrete and the continuous description,, Phys. A, 381 (2007), 39. doi: 10.1016/j.physa.2007.03.045. Google Scholar
[17]	M. E. Taylor, Partial Differential Equations III,, Second edition. Applied Mathematical Sciences, (2011). doi: 10.1007/978-1-4419-7049-7. Google Scholar
[18]	Y. Tu, J. Toner and M. Ulm, Sound waves and the absence of Galilean invariance in flocks,, Phys. Rev. Lett., 80 (1998), 4819. doi: 10.1103/PhysRevLett.80.4819. Google Scholar
[19]	T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles,, Phys. Rev. Lett., 75 (1995), 1226. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

[1]	Xuemei Li, Zaijiu Shang. On the existence of invariant tori in non-conservative dynamical systems with degeneracy and finite differentiability. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4225-4257. doi: 10.3934/dcds.2019171
[2]	Song Wang. Numerical solution of an obstacle problem with interval coefficients. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019030
[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]	Roberto Garrappa, Eleonora Messina, Antonia Vecchio. Effect of perturbation in the numerical solution of fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2679-2694. doi: 10.3934/dcdsb.2017188
[5]	María Teresa Cao-Rial, Peregrina Quintela, Carlos Moreno. Numerical solution of a time-dependent Signorini contact problem. Conference Publications, 2007, 2007 (Special) : 201-211. doi: 10.3934/proc.2007.2007.201
[6]	Alexandre Caboussat, Roland Glowinski. Numerical solution of a variational problem arising in stress analysis: The vector case. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1447-1472. doi: 10.3934/dcds.2010.27.1447
[7]	Steinar Evje, Huanyao Wen, Lei Yao. Global solutions to a one-dimensional non-conservative two-phase model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1927-1955. doi: 10.3934/dcds.2016.36.1927
[8]	Prasanta Kumar Barik, Ankik Kumar Giri. A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation. Kinetic & Related Models, 2018, 11 (5) : 1125-1138. doi: 10.3934/krm.2018043
[9]	Jin Lai, Huanyao Wen, Lei Yao. Vanishing capillarity limit of the non-conservative compressible two-fluid model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1361-1392. doi: 10.3934/dcdsb.2017066
[10]	G. Bellettini, Giorgio Fusco, Nicola Guglielmi. A concept of solution and numerical experiments for forward-backward diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 783-842. doi: 10.3934/dcds.2006.16.783
[11]	Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283
[12]	Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2019061
[13]	Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[14]	Chunqing Lu. Existence and uniqueness of single spike solution of the carrier-pearson problem. Conference Publications, 2001, 2001 (Special) : 259-264. doi: 10.3934/proc.2001.2001.259
[15]	Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033
[16]	Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure & Applied Analysis, 2005, 4 (3) : 487-498. doi: 10.3934/cpaa.2005.4.487
[17]	Tong Li, Anthony Suen. Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 861-875. doi: 10.3934/dcds.2016.36.861
[18]	Dominique Blanchard, Olivier Guibé, Hicham Redwane. Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 197-217. doi: 10.3934/cpaa.2016.15.197
[19]	Roman Chapko, B. Tomas Johansson. On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach. Inverse Problems & Imaging, 2012, 6 (1) : 25-38. doi: 10.3934/ipi.2012.6.25
[20]	Ansgar Jüngel, Josipa-Pina Milišić. Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution. Kinetic & Related Models, 2011, 4 (3) : 785-807. doi: 10.3934/krm.2011.4.785

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences