American Institute of Mathematical Sciences

Advanced
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

Benjamin Jourdain 1, and  Julien Reygner 2,

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.
Keywords: Multitype sticky particle dynamics, rate of convergence., hyperbolic systems.
Mathematics Subject Classification: 35L45, 65M12, 82C2.
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 & Continuous Dynamical Systems - A, 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.  doi: 10.1016/0020-0190(79)90072-3.  Google Scholar

[2]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems,, Ann. of Math. (2), 161 (2005), 223.  doi: 10.4007/annals.2005.161.223.  Google Scholar

[3]

S. Bobkov and M. Ledoux, One dimensional empirical measures, order statistics, and Kantorovich transport distances,, preprint, ().   Google Scholar

[4]

F. Bouchut, On Zero Pressure Gas Dynamics,, in Series on Advances in Mathematics for Applied Sciences, 22 (1994), 171.   Google Scholar

[5]

F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness,, Comm. Partial Differential Equations, 24 (1999), 2173.  doi: 10.1080/03605309908821498.  Google Scholar

[6]

Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws,, SIAM J. Numer. Anal., 35 (1998), 2317.  doi: 10.1137/S0036142997317353.  Google Scholar

[7]

A. Bressan and T. Nguyen, Non-existence and non-uniqueness for multidimensional sticky particle systems,, Kinet. Relat. Models, 7 (2014), 205.  doi: 10.3934/krm.2014.7.205.  Google Scholar

[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.  doi: 10.1007/BF02101897.  Google Scholar

[9]

M. T. Goodrich, Finding the convex hull of a sorted point set in parallel,, Inform. Process. Lett., 26 (1987), 173.  doi: 10.1016/0020-0190(87)90002-0.  Google Scholar

[10]

R. L. Graham, An efficient algorithm for determining the convex hull of a finite planar set,, Inform. Process. Lett., 1 (1972), 132.   Google Scholar

[11]

E. Grenier, Existence globale pour le système des gaz sans pression,, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 171.   Google Scholar

[12]

B. Jourdain, Signed sticky particles and 1D scalar conservation laws,, C. R. Math. Acad. Sci. Paris, 334 (2002), 233.  doi: 10.1016/S1631-073X(02)02251-3.  Google Scholar

[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, ().   Google Scholar

[14]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts Appl. Math.,, Cambridge University Press, (2002).  doi: 10.1017/CBO9780511791253.  Google Scholar

[15]

D. Serre, Systems of Conservation Laws I,, Cambridge University Press, (1999).  doi: 10.1017/CBO9780511612374.  Google Scholar

[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.   Google Scholar

[17]

C. Villani, Optimal Transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],, Springer-Verlag, (2009).  doi: 10.1007/978-3-540-71050-9.  Google Scholar

[18]

Y. B. Zel'dovitch, Gravitational instability: An approximate theory for large density perturbations,, Astron. Astroph., 5 (1970), 84.   Google Scholar

show all references

References:
[1]

A. M. Andrew, Another efficient algorithm for convex hulls in two dimensions,, Inform. Process. Lett., 9 (1979), 216.  doi: 10.1016/0020-0190(79)90072-3.  Google Scholar

[2]

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems,, Ann. of Math. (2), 161 (2005), 223.  doi: 10.4007/annals.2005.161.223.  Google Scholar

[3]

S. Bobkov and M. Ledoux, One dimensional empirical measures, order statistics, and Kantorovich transport distances,, preprint, ().   Google Scholar

[4]

F. Bouchut, On Zero Pressure Gas Dynamics,, in Series on Advances in Mathematics for Applied Sciences, 22 (1994), 171.   Google Scholar

[5]

F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness,, Comm. Partial Differential Equations, 24 (1999), 2173.  doi: 10.1080/03605309908821498.  Google Scholar

[6]

Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws,, SIAM J. Numer. Anal., 35 (1998), 2317.  doi: 10.1137/S0036142997317353.  Google Scholar

[7]

A. Bressan and T. Nguyen, Non-existence and non-uniqueness for multidimensional sticky particle systems,, Kinet. Relat. Models, 7 (2014), 205.  doi: 10.3934/krm.2014.7.205.  Google Scholar

[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.  doi: 10.1007/BF02101897.  Google Scholar

[9]

M. T. Goodrich, Finding the convex hull of a sorted point set in parallel,, Inform. Process. Lett., 26 (1987), 173.  doi: 10.1016/0020-0190(87)90002-0.  Google Scholar

[10]

R. L. Graham, An efficient algorithm for determining the convex hull of a finite planar set,, Inform. Process. Lett., 1 (1972), 132.   Google Scholar

[11]

E. Grenier, Existence globale pour le système des gaz sans pression,, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 171.   Google Scholar

[12]

B. Jourdain, Signed sticky particles and 1D scalar conservation laws,, C. R. Math. Acad. Sci. Paris, 334 (2002), 233.  doi: 10.1016/S1631-073X(02)02251-3.  Google Scholar

[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, ().   Google Scholar

[14]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts Appl. Math.,, Cambridge University Press, (2002).  doi: 10.1017/CBO9780511791253.  Google Scholar

[15]

D. Serre, Systems of Conservation Laws I,, Cambridge University Press, (1999).  doi: 10.1017/CBO9780511612374.  Google Scholar

[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.   Google Scholar

[17]

C. Villani, Optimal Transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],, Springer-Verlag, (2009).  doi: 10.1007/978-3-540-71050-9.  Google Scholar

[18]

Y. B. Zel'dovitch, Gravitational instability: An approximate theory for large density perturbations,, Astron. Astroph., 5 (1970), 84.   Google Scholar

[1]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020210
[2]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513
[3]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166
[4]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011
[5]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225
[6]

Simone Calogero, Juan Calvo, Óscar Sánchez, Juan Soler. Dispersive behavior in galactic dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 1-16. doi: 10.3934/dcdsb.2010.14.1
[7]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[8]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[9]

Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053
[10]

Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29
[11]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133
[12]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008
[13]

Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53
[14]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002
[15]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185
[16]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185
[17]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[18]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301
[19]

Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109
[20]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences