Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data

Benjamin  Jourdain; Julien  Reygner

doi:10.3934/dcds.2016015

Article Contents

2016, Volume 36, Issue 9: 4963-4996. Doi: 10.3934/dcds.2016015

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Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data

Benjamin Jourdain^1, and
Julien Reygner^2,

1.
CERMICS, École des Ponts, UPE, Inria, Champs-sur-Marne
2.
CERMICS, École des Ponts, UPE, Champs-sur-Marne

Received: June 30, 2015

Revised: December 31, 2015

Published: May 2017

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Abstract

Brenier and Grenier [SIAM J. Numer. Anal., 1998] proved that sticky particle dynamics with a large number of particles allow to approximate the entropy solution to scalar one-dimensional conservation laws with monotonic initial data. In [arXiv:1501.01498], we introduced a multitype version of this dynamics and proved that the associated empirical cumulative distribution functions converge to the viscosity solution, in the sense of Bianchini and Bressan [Ann. of Math. (2), 2005], of one-dimensional diagonal hyperbolic systems with monotonic initial data of arbitrary finite variation. In the present paper, we analyse the $L^1$ error of this approximation procedure, by splitting it into the discretisation error of the initial data and the non-entropicity error induced by the evolution of the particle system. We prove that the error at time $t$ is bounded from above by a term of order $(1+t)/n$, where $n$ denotes the number of particles, and give an example showing that this rate is optimal. We last analyse the additional error introduced when replacing the multitype sticky particle dynamics by an iterative scheme based on the typewise sticky particle dynamics, and illustrate the convergence of this scheme by numerical simulations.

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Mathematics Subject Classification: 35L45, 65M12, 82C21.

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