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Interplay Between Diffusion Anisotropy and Mesh Skewness in Hybrid High-Order Schemes Read chapter Read first chapter

2020 | OriginalPaper | Chapter

Finite Volume Method for a System of Continuity Equations Driven by Nonlocal Interactions

Authors : Anissa El Keurti, Thomas Rey

Published in: Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples

Publisher: Springer International Publishing

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Abstract

We present a new finite volume method for computing numerical approximations of a system of nonlocal transport equation modeling interacting species. This method is based on the work [F. Delarue, F. Lagoutière, N. Vauchelet, Convergence analysis of upwind type schemes for the aggregation equation with pointy potential, Ann. Henri. Lebesgue 2019], where the nonlocal continuity equations are treated as conservative transport equations with a nonlocal, nonlinear, rough velocity field. We analyze some properties of the method, and illustrate the results with numerical simulations.

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previous chapter A Finite-Volume Scheme for a Cross-Diffusion Model Arising from Interacting Many-Particle Population Systems
next chapter A Macroscopic Model to Reproduce Self-organization at Bottlenecks
Literature
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Bertozzi, A.L., Carrillo, J.A., Laurent, T.: Blow-up in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22(3), 683 (2009)MathSciNetCrossRef
2.
Carrillo, J.A., Francesco, M.D., Esposito, A., Fagioli, S., Schmidtchen, M.: Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions. Discr. Cont. Dyn. Sys. A 40, 1191 (2020)MathSciNetCrossRef
3.
Carrillo, J.A., James, F., Lagoutière, F., Vauchelet, N.: The filippov characteristic flow for the aggregation equation with mildly singular potentials. J. Diff. Eq. 260(1), 304–338 (2016)MathSciNetCrossRef
4.
Delarue, F., Lagoutière, F., Vauchelet, N.: Convergence order of upwind type schemes for transport equations with discontinuous coefficients. J. Math. Pures. App. 108(6), 918–951 (2017)MathSciNetCrossRef
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Di Francesco, M., Fagioli, S.: Measure solutions for non-local interaction pdes with two species. Nonlinearity 26(10), 2777 (2013)MathSciNetCrossRef
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Di Francesco, M., Fagioli, S.: A nonlocal swarm model for predators-prey interactions. Math. Mod. Meth. App. Sci. 26(02), 319–355 (2016)MathSciNetCrossRef
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Emako-Kazianou, C., Liao, J., Vauchelet, N.: Synchronising and non-synchronising dynamics for a two-species aggregation model. Discr. Cont. Dyn. Sys. B 22(6), 2121–2146 (2017)MathSciNetMATH
Metadata
Title
Finite Volume Method for a System of Continuity Equations Driven by Nonlocal Interactions
Authors
Anissa El Keurti
Thomas Rey
Copyright Year
2020
Book
Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples

Print ISBN: 978-3-030-43650-6

Electronic ISBN: 978-3-030-43651-3

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-43651-3_20

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