Geodesic
On the Numerical Integration of Scalar Nonlocal Conservation Laws
Amorim, Paulo1 ; Colombo, Rinaldo M.2  ; Teixeira, Andreia3
1 Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, Cidade Universitária 21945–970, Rio de Janeiro, Brazil.
2 Unità INdAM, Università di Brescia, Via Branze 38, 25123 Brescia, Italy.
3 Centro de Matemática e Aplicações Fundamentais, Departamento de Matemática, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 19-37

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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.

DOI : 10.1051/m2an/2014023
Classification : 35L65
Keywords: Nonlocal conservation laws, Lax Friedrichs scheme

Amorim, Paulo 1 ; Colombo, Rinaldo M. 2 ; Teixeira, Andreia 3

1 Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, Cidade Universitária 21945–970, Rio de Janeiro, Brazil.
2 Unità INdAM, Università di Brescia, Via Branze 38, 25123 Brescia, Italy.
3 Centro de Matemática e Aplicações Fundamentais, Departamento de Matemática, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal. 
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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

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