Abstract
We present the numerical solution of a hydrodynamics model of flocking using a suitable modified semi-implicit discontinuous Galerkin method. The investigated model describing the dynamics of flocks of birds or other individual entities forming herds or swarms was introduced by Fornasier et al. (Physica D 240(1):21–31, 2011). The main idea of this model comes from the well known Cucker-Smale model. The resulting equations consist of the Euler equations for compressible flow with an additional non-local non-linear source term. The model is discretized by the semi-implicit discontinuous Galerkin method for the compressible Euler equations of Feistauer and Kučera (J Comput Phys 224(1):208–221, 2007). We show that with a suitable treatment of the source term we can use this method for models like the model of flocking and find a numerical solution very efficiently.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
T.A. Davis, I.S. Duff, A combined unifrontal/multifrontal method for unsymmetric sparse matrices. ACM Trans. Math. Softw. 25, 1–19 (1999)
M. Feistauer, V. Kučera, On a robust discontinuous Galerkin technique for the solution of compressible flow. J. Comput. Phys. 224, 208–221 (2007)
M. Feistauer, J. Felcman, I. Straškraba, Mathematical and Computational Methods for Compressible Flow (Clarendon Press, Oxford, 2003)
M. Fornasier, J. Haškovec, G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation. Physica D 240 (1), 21–31 (2011)
G. Vijayasundaram, Transonic flow simulation using upstream centered scheme of Godunov type in finite elements. J. Comput. Phys. 63, 416–433 (1986)
Acknowledgements
The research of V. Kučera is supported by the Grant No. P201/13/00522S of the Czech Science Foundation. He is currently a Fulbright visiting scholar at Brown University, Providence, RI, USA, supported by the J. William Fulbright Commission in the Czech Republic. The research of A. Živčáková is supported by the Charles University in Prague, project GA UK No. 758214.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Živčáková, A., Kučera, V. (2016). Semi-implicit DGM Applied to a Model of Flocking. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-39929-4_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-39927-0
Online ISBN: 978-3-319-39929-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)