We study a rather general class of 1D nonlocal conservation laws from a numerical point of view. First, following [F. Betancourt, R. Bürger, K.H. Karlsen and E.M. Tory, On nonlocal conservation laws modelling sedimentation. Nonlinearity 24 (2011) 855–885], we define an algorithm to numerically integrate them and prove its convergence. Then, we use this algorithm to investigate various analytical properties, obtaining evidence that usual properties of standard conservation laws fail in the nonlocal setting. Moreover, on the basis of our numerical integrations, we are led to conjecture the convergence of the nonlocal equation to the local ones, although no analytical results are, to our knowledge, available in this context.
Mots clés : Nonlocal conservation laws, Lax Friedrichs scheme
@article{M2AN_2015__49_1_19_0,
author = {Amorim, Paulo and Colombo, Rinaldo M. and Teixeira, Andreia},
title = {On the {Numerical} {Integration} of {Scalar} {Nonlocal} {Conservation} {Laws}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {19--37},
publisher = {EDP-Sciences},
volume = {49},
number = {1},
year = {2015},
doi = {10.1051/m2an/2014023},
mrnumber = {3342191},
zbl = {1317.65165},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2014023/}
}
TY - JOUR AU - Amorim, Paulo AU - Colombo, Rinaldo M. AU - Teixeira, Andreia TI - On the Numerical Integration of Scalar Nonlocal Conservation Laws JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 19 EP - 37 VL - 49 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2014023/ DO - 10.1051/m2an/2014023 LA - en ID - M2AN_2015__49_1_19_0 ER -
%0 Journal Article %A Amorim, Paulo %A Colombo, Rinaldo M. %A Teixeira, Andreia %T On the Numerical Integration of Scalar Nonlocal Conservation Laws %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 19-37 %V 49 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2014023/ %R 10.1051/m2an/2014023 %G en %F M2AN_2015__49_1_19_0
Amorim, Paulo; Colombo, Rinaldo M.; Teixeira, Andreia. On the Numerical Integration of Scalar Nonlocal Conservation Laws. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 19-37. doi : 10.1051/m2an/2014023. http://www.numdam.org/articles/10.1051/m2an/2014023/
, , and , Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math. 64 (2003) 41–80. | DOI | MR | Zbl
, , and , On nonlocal conservation laws modelling sedimentation. Nonlinearity 24 (2011) 855–885. | DOI | MR | Zbl
A. Bressan, Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem, vol. 20 of Oxford Lect. Ser. Math. Applications. Oxford University Press, Oxford (2000). | MR | Zbl
, The linearization of a boundary value problem for a scalar conservation law. Commun. Math. Sci. 6 (2008) 651–667. | DOI | MR | Zbl
, , and , Structured populations, cell growth and measure valued balance laws. J. Differ. Eqs. 252 (2012) 3245–3277. | DOI | MR | Zbl
, and , A class of nonlocal models for pedestrian traffic. Math. Models Methods Appl. Sci. 22 (2012) 1150023, 34. | DOI | MR | Zbl
, and , Control of the continuity equation with a non local flow. ESAIM: COCV 17 (2011) 353–379. | Numdam | MR | Zbl
and , Nonlocal Crowd Dynamics Models for Several Populations. Acta Math. Sci. Ser. B Engl. Ed. 32 (2012) 177–196. | DOI | MR | Zbl
, and , Stability and total variation estimates on general scalar balance laws. Commun. Math. Sci. 7 (2009) 37–65. | DOI | MR | Zbl
G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow. Nonlinear Differ. Eqs. Appl. NoDEA (2012) 1–15. | MR | Zbl
C.M. Dafermos, Hyperbolic conservation laws in continuum physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 3rd edition (2010). | MR | Zbl
, Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review. J. Math. Biol. 65 (2012) 35–75. | DOI | MR | Zbl
, Conservation laws in mathematical biology. Discrete Contin. Dyn. Syst. 32 (2012) 3081–3097. | DOI | MR | Zbl
and . Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients. Math. Comput. 69 987–1015, 2000. | DOI | MR | Zbl
, and , A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients. J. Differ. Eqs. 248 (2010) 2703–2735. | DOI | MR | Zbl
S. Gttlich, S. Hoher, P. Schindler, V. Schleper and A. Verl, Modeling, simulation and validation of material flow on conveyor belts. Prepint Reihe des Stuttgart Research Center for Simulation Technology, 746 (2013).
and , Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. ESAIM: M2AN 35 (2001) 239–269. | DOI | Numdam | MR | Zbl
, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) (123) 228–255. | MR | Zbl
R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002). | MR | Zbl
and , On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London Ser. A. 229 (1955) 317–345. | DOI | MR | Zbl
, and , High-field limit for the Vlasov-Poisson-Fokker-Planck system. Arch. Ration. Mech. Anal. 158 (2001) 29–59. | DOI | MR | Zbl
and , Time-evolving measures and macroscopic modeling of pedestrian flow. Arch. Ration. Mech. Anal. 199 (2011) 707–738. | DOI | MR | Zbl
, Shock waves on the highway. Oper. Res. 4 (1956) 42–51. | DOI | MR | Zbl
Cité par Sources :
