# American Institute of Mathematical Sciences

September  2016, 36(9): 4963-4996. doi: 10.3934/dcds.2016015

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

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

Received  July 2015 Revised  January 2016 Published  May 2016

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.
Citation: Benjamin Jourdain, Julien Reygner. Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4963-4996. doi: 10.3934/dcds.2016015
##### References:
 [1] A. M. Andrew, Another efficient algorithm for convex hulls in two dimensions, Inform. Process. Lett., 9 (1979), 216-219. doi: 10.1016/0020-0190(79)90072-3. [2] S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2), 161 (2005), 223-342. doi: 10.4007/annals.2005.161.223. [3] S. Bobkov and M. Ledoux, One dimensional empirical measures, order statistics, and Kantorovich transport distances,, preprint, (). [4] F. Bouchut, On Zero Pressure Gas Dynamics, in Series on Advances in Mathematics for Applied Sciences, World Scientific, 22 (1994), 171-190. [5] F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations, 24 (1999), 2173-2189. doi: 10.1080/03605309908821498. [6] Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: 10.1137/S0036142997317353. [7] A. Bressan and T. Nguyen, Non-existence and non-uniqueness for multidimensional sticky particle systems, Kinet. Relat. Models, 7 (2014), 205-218. doi: 10.3934/krm.2014.7.205. [8] W. E, Y. G. Rykov and Y. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys., 177 (1996), 349-380. doi: 10.1007/BF02101897. [9] M. T. Goodrich, Finding the convex hull of a sorted point set in parallel, Inform. Process. Lett., 26 (1987), 173-179. doi: 10.1016/0020-0190(87)90002-0. [10] R. L. Graham, An efficient algorithm for determining the convex hull of a finite planar set, Inform. Process. Lett., 1 (1972), 132-133. [11] E. Grenier, Existence globale pour le système des gaz sans pression, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 171-174. [12] B. Jourdain, Signed sticky particles and 1D scalar conservation laws, C. R. Math. Acad. Sci. Paris, 334 (2002), 233-238. doi: 10.1016/S1631-073X(02)02251-3. [13] B. Jourdain and J. Reygner, A multitype sticky particle construction of Wasserstein stable semigroups solving one-dimensional diagonal hyperbolic systems with large monotonic data,, preprint, (). [14] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [15] D. Serre, Systems of Conservation Laws I, Cambridge University Press, Cambridge, 1999. Translated from the 1996 French original by I. N. Sneddon. doi: 10.1017/CBO9780511612374. [16] M. Vergassola, B. Dubrulle, U. Frisch and A. Noullez, Burgers' equation, devil's staircases and the mass distribution for large-scale structures, Astron. Astroph., 289 (1994), 325-356. [17] C. Villani, Optimal Transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. [18] Y. B. Zel'dovitch, Gravitational instability: An approximate theory for large density perturbations, Astron. Astroph., 5 (1970), 84-89.

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##### References:
 [1] A. M. Andrew, Another efficient algorithm for convex hulls in two dimensions, Inform. Process. Lett., 9 (1979), 216-219. doi: 10.1016/0020-0190(79)90072-3. [2] S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2), 161 (2005), 223-342. doi: 10.4007/annals.2005.161.223. [3] S. Bobkov and M. Ledoux, One dimensional empirical measures, order statistics, and Kantorovich transport distances,, preprint, (). [4] F. Bouchut, On Zero Pressure Gas Dynamics, in Series on Advances in Mathematics for Applied Sciences, World Scientific, 22 (1994), 171-190. [5] F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations, 24 (1999), 2173-2189. doi: 10.1080/03605309908821498. [6] Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: 10.1137/S0036142997317353. [7] A. Bressan and T. Nguyen, Non-existence and non-uniqueness for multidimensional sticky particle systems, Kinet. Relat. Models, 7 (2014), 205-218. doi: 10.3934/krm.2014.7.205. [8] W. E, Y. G. Rykov and Y. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys., 177 (1996), 349-380. doi: 10.1007/BF02101897. [9] M. T. Goodrich, Finding the convex hull of a sorted point set in parallel, Inform. Process. Lett., 26 (1987), 173-179. doi: 10.1016/0020-0190(87)90002-0. [10] R. L. Graham, An efficient algorithm for determining the convex hull of a finite planar set, Inform. Process. Lett., 1 (1972), 132-133. [11] E. Grenier, Existence globale pour le système des gaz sans pression, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 171-174. [12] B. Jourdain, Signed sticky particles and 1D scalar conservation laws, C. R. Math. Acad. Sci. Paris, 334 (2002), 233-238. doi: 10.1016/S1631-073X(02)02251-3. [13] B. Jourdain and J. Reygner, A multitype sticky particle construction of Wasserstein stable semigroups solving one-dimensional diagonal hyperbolic systems with large monotonic data,, preprint, (). [14] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [15] D. Serre, Systems of Conservation Laws I, Cambridge University Press, Cambridge, 1999. Translated from the 1996 French original by I. N. Sneddon. doi: 10.1017/CBO9780511612374. [16] M. Vergassola, B. Dubrulle, U. Frisch and A. Noullez, Burgers' equation, devil's staircases and the mass distribution for large-scale structures, Astron. Astroph., 289 (1994), 325-356. [17] C. Villani, Optimal Transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. [18] Y. B. Zel'dovitch, Gravitational instability: An approximate theory for large density perturbations, Astron. Astroph., 5 (1970), 84-89.
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