publication . Article . Preprint . 2014

A discontinuous Galerkin method on kinetic flocking models

Changhui Tan;
Open Access
  • Published: 18 Sep 2014 Journal: Mathematical Models and Methods in Applied Sciences, volume 27, pages 1,199-1,221 (issn: 0218-2025, eissn: 1793-6314, Copyright policy)
  • Publisher: World Scientific Pub Co Pte Lt
Abstract
<jats:p> We study kinetic representations of flocking models. They arise from agent-based models for self-organized dynamics, such as Cucker–Smale [Emergent behaviors in flocks, IEEE Trans. Autom. Control. 52 (2007) 852–862] and Motsch–Tadmor [A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys. 144 (2011) 923–947] models. We first establish a well-posedness theory and large-time flocking behavior for the kinetic systems, which indicates a concentration in velocity variable in infinite time. We then apply a discontinuous Galerkin method to treat the asymptotic [Formula: see text]-singularity, and construct high-order positive-pres...
Persistent Identifiers
Subjects
free text keywords: Modelling and Simulation, Applied Mathematics, Mathematics - Numerical Analysis, 65M60, 92C45, Control theory, Flocking (texture), Flocking (behavior), Mathematics, Discontinuous Galerkin method, Kinetic energy
18 references, page 1 of 2

[1] A. L. BERTOZZI, J. A. CARRILLO AND T. LAURENT, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity 22, no. 3, (2009): 683-710.

[2] J. A. CARRILLO, M. FORNASIER, J. ROSADO AND G. TOSCANI, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM Journal on Mathematical Analysis, 42, no. 1, (2010): 218-236.

[3] G.-Q. CHEN AND H. LIU, Formation of δ -shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM journal on mathematical analysis 34, no. 4, (2003): 925-938.

[4] B. COCKBURN AND C.-W. SHU, TVB Runge-Kutta Local projection discontinuous Galerkin finite element method for conservation law II: General framework, Mathematics of Computation, 52, (1989): 411-435.

[5] F. CUCKER AND S. SMALE, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52, no. 5, (2007): 852-862.

[6] G. DIMARCO AND L. PARESCHI, Numerical methods for kinetic equations, Acta Numerica, 23 (2014): 369-520. [OpenAIRE]

[7] S. GOTTLIEB, C.-W. SHU AND E. TADMOR, Strong stability preserving high-order time discretization methods, SIAM Review, 43, (2001): 89-112.

[8] S.-Y. HA AND J.-G. LIU, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7, no. 2, (2009): 297-325.

[9] S.-Y. HA AND E. TADMOR, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1, no. 3, (2008): 415-435. [OpenAIRE]

[10] R. MCLACHLAN AND G. QUISPEL, Splitting methods, Acta Numerica 11.0 (2002): 341-434.

[11] S. MOTSCH AND E. TADMOR, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys, 144(5) (2011) 923-947.

[12] W.H. REED AND T.R. HILL, Triangular mesh methods for the Neutron transport equation, Los Alamos Scientific Laboratory Report LA-UR-73-479, Los Alamos, NM, 1973.

[13] C.-W. SHU, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Springer Berlin Heidelberg, 1998.

[14] E. TADMOR AND C. TAN, Critical thresholds in flocking hydrodynamics with nonlocal alignment, to appear at Phil. Trans. R. Soc. A.

[15] Y. YANG AND C.-W. SHU, Discontinuous Galerkin method for hyperbolic equations involving δ -singularities: negative-order norm error estimates and applications, Numerische Mathematik, (2013): 1-29.

18 references, page 1 of 2
Abstract
<jats:p> We study kinetic representations of flocking models. They arise from agent-based models for self-organized dynamics, such as Cucker–Smale [Emergent behaviors in flocks, IEEE Trans. Autom. Control. 52 (2007) 852–862] and Motsch–Tadmor [A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys. 144 (2011) 923–947] models. We first establish a well-posedness theory and large-time flocking behavior for the kinetic systems, which indicates a concentration in velocity variable in infinite time. We then apply a discontinuous Galerkin method to treat the asymptotic [Formula: see text]-singularity, and construct high-order positive-pres...
Persistent Identifiers
Subjects
free text keywords: Modelling and Simulation, Applied Mathematics, Mathematics - Numerical Analysis, 65M60, 92C45, Control theory, Flocking (texture), Flocking (behavior), Mathematics, Discontinuous Galerkin method, Kinetic energy
18 references, page 1 of 2

[1] A. L. BERTOZZI, J. A. CARRILLO AND T. LAURENT, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity 22, no. 3, (2009): 683-710.

[2] J. A. CARRILLO, M. FORNASIER, J. ROSADO AND G. TOSCANI, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM Journal on Mathematical Analysis, 42, no. 1, (2010): 218-236.

[3] G.-Q. CHEN AND H. LIU, Formation of δ -shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM journal on mathematical analysis 34, no. 4, (2003): 925-938.

[4] B. COCKBURN AND C.-W. SHU, TVB Runge-Kutta Local projection discontinuous Galerkin finite element method for conservation law II: General framework, Mathematics of Computation, 52, (1989): 411-435.

[5] F. CUCKER AND S. SMALE, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52, no. 5, (2007): 852-862.

[6] G. DIMARCO AND L. PARESCHI, Numerical methods for kinetic equations, Acta Numerica, 23 (2014): 369-520. [OpenAIRE]

[7] S. GOTTLIEB, C.-W. SHU AND E. TADMOR, Strong stability preserving high-order time discretization methods, SIAM Review, 43, (2001): 89-112.

[8] S.-Y. HA AND J.-G. LIU, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7, no. 2, (2009): 297-325.

[9] S.-Y. HA AND E. TADMOR, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1, no. 3, (2008): 415-435. [OpenAIRE]

[10] R. MCLACHLAN AND G. QUISPEL, Splitting methods, Acta Numerica 11.0 (2002): 341-434.

[11] S. MOTSCH AND E. TADMOR, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys, 144(5) (2011) 923-947.

[12] W.H. REED AND T.R. HILL, Triangular mesh methods for the Neutron transport equation, Los Alamos Scientific Laboratory Report LA-UR-73-479, Los Alamos, NM, 1973.

[13] C.-W. SHU, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Springer Berlin Heidelberg, 1998.

[14] E. TADMOR AND C. TAN, Critical thresholds in flocking hydrodynamics with nonlocal alignment, to appear at Phil. Trans. R. Soc. A.

[15] Y. YANG AND C.-W. SHU, Discontinuous Galerkin method for hyperbolic equations involving δ -singularities: negative-order norm error estimates and applications, Numerische Mathematik, (2013): 1-29.

18 references, page 1 of 2
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