Consistency, accuracy and entropy behaviour of remeshed particle methods
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, pp. 57-81.

In this paper we analyze the consistency, the accuracy and some entropy properties of particle methods with remeshing in the case of a scalar one-dimensional conservation law. As in [G.-H. Cottet and L. Weynans, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 51-56] we re-write particle methods with remeshing in the finite-difference formalism. This allows us to prove the consistency of these methods, and accuracy properties related to the accuracy of interpolation kernels. Cottet and Magni devised recently in [G.-H. Cottet and A. Magni, C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1367-1372] and [A. Magni and G.-H. Cottet, J. Comput. Phys. 231 (2012) 152-172] TVD remeshing schemes for particle methods. We extend these results to the nonlinear case with arbitrary velocity sign. We present numerical results obtained with these new TVD particle methods for the Euler equations in the case of the Sod shock tube. Then we prove that with these new TVD remeshing schemes the particle methods converge toward the entropy solution of the scalar conservation law.

DOI: 10.1051/m2an/2012019
Classification: 65M12,  65M75
Keywords: particle methods with remeshing, interpolation kernels, consistency, truncation error, entropy inequalities, total variation, limiters, convergence
@article{M2AN_2013__47_1_57_0,
author = {Weynans, Lisl and Magni, Adrien},
title = {Consistency, accuracy and entropy behaviour of remeshed particle methods},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {57--81},
publisher = {EDP-Sciences},
volume = {47},
number = {1},
year = {2013},
doi = {10.1051/m2an/2012019},
zbl = {1278.65136},
mrnumber = {2968695},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2012019/}
}
TY  - JOUR
AU  - Weynans, Lisl
TI  - Consistency, accuracy and entropy behaviour of remeshed particle methods
JO  - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY  - 2013
DA  - 2013///
SP  - 57
EP  - 81
VL  - 47
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2012019/
UR  - https://zbmath.org/?q=an%3A1278.65136
UR  - https://www.ams.org/mathscinet-getitem?mr=2968695
UR  - https://doi.org/10.1051/m2an/2012019
DO  - 10.1051/m2an/2012019
LA  - en
ID  - M2AN_2013__47_1_57_0
ER  - 
%0 Journal Article
%A Weynans, Lisl
%T Consistency, accuracy and entropy behaviour of remeshed particle methods
%J ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
%D 2013
%P 57-81
%V 47
%N 1
%I EDP-Sciences
%U https://doi.org/10.1051/m2an/2012019
%R 10.1051/m2an/2012019
%G en
%F M2AN_2013__47_1_57_0
Weynans, Lisl; Magni, Adrien. Consistency, accuracy and entropy behaviour of remeshed particle methods. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, pp. 57-81. doi : 10.1051/m2an/2012019. http://www.numdam.org/articles/10.1051/m2an/2012019/

[1] B. Ben Moussa and J.P. Vila, Convergence of SPH methods for scalar nonlinear conservation laws. SIAM J. Numer. Anal. 37 (2000) 863-887. | MR | Zbl

[2] W. Benz, The Numerical Modelling of Nonlinear Stellar Pulsations, Problems and Prospects, a review, in Smooth Particle Hydrodynamics : NATO ASIS Series (1989) 269-287.

[3] C. Berthon, Contribution à l'analyse numérique des équations de Navier-Stokes compressibles à deux entropies spécifiques. Application à la turbulence compressible. Ph.D. thesis, Université Paris VI (1998).

[4] M. Coquerelle and G.-H. Cottet, A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies. J. Comput. Phys. 227 (2008) 9121-9137. | MR | Zbl

[5] G.-H. Cottet and P.D. Koumoutsakos, Vortex methods. Cambridge University Press (2000). | MR | Zbl

[6] G.-H. Cottet and A. Magni, TVD remeshing schemes for particle methods. C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1367-1372. | MR | Zbl

[7] G.-H. Cottet and L. Weynans, Particle methods revisited : a class of high-order finite-difference schemes. C. R. Acad. Sci. Paris, Ser. I 343 (2006) 51-56. | MR | Zbl

[8] G.-H. Cottet, B. Michaux, S. Ossia and G. Vanderlinden, A comparison of spectral and vortex methods in three-dimensional incompressible flow. J. Comput. Phys. 175 (2002) 702-712. | Zbl

[9] M.W. Evans and F.H. Harlow, The particle-in-cell method for hydrodynamics calculations. Technical Report, Los Alamos Scientific Laboratory (1956).

[10] A. Ghoniem and D. Wee, Modified interpolation kernels for treating diffusion and remeshing in vortex methods. J. Comput. Phys. 213 (2006) 239-263. | MR | Zbl

[11] R.A. Gingold and J.J. Monaghan, Smoothed particle hydrodynamics : theory and application to non-spherical stars. Mon. Not. R. Astron. Soc. 181 (1977) 375-389. | Zbl

[12] F.H. Harlow, Hydrodynamic problems involving large fluid distorsion. J. Assoc. Comput. Mach. 4 (1957) 137-142.

[13] A. Harten, High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357-393. | MR | Zbl

[14] T. Hou and P.G. Lefloch, Why non-conservative schemes converge to wrong solutions : error analysis. Math. Comput. 62 (1994) 497-530. | MR | Zbl

[15] P. Koumoutsakos and S. Hieber. A Lagrangian particle level set method. J. Comput. Phys. 210 (2005) 342-367. | MR | Zbl

[16] P. Koumoutsakos and A. Leonard, High resolution simulations of the flow around an impulsively started cylinder using vortex methods. J. Fluid Mech. 296 (1995) 1-38. | Zbl

[17] N. Lanson and J.P. Vila, Convergence des méthodes particulaires renormalisées pour les systèmes de Friedrichs. C. R. Acad. Sci. Paris, Ser. I 349 (2005) 465-470. | MR | Zbl

[18] N. Lanson and J.P. Vila, Renormalized meshfree schemes II : convergence for scalar conservation laws. SIAM J. Numer. Anal. 46 (2008) 1935-1964. | MR | Zbl

[19] R.J. Leveque, Finite-volume methods for hyperbolic problems. Cambridge University Press (2002). | MR | Zbl

[20] A. Magni, Méthodes particulaires avec remaillage : analyse numérique nouveaux schémas et applications pour la simulation d'équations de transport. Ph.D. thesis, Université de Grenoble. Available on : http://tel.archives-ouvertes.fr/ tel-00623128/fr/ (2011).

[21] A. Magni and G.-H. Cottet, Accurate, non-oscillatory, remeshing schemes for particle methods. J. Comput. Phys. 231 (2012) 152-172. | MR

[22] A. Majda and S. Osher, Numerical viscosity and the entropy condition. Commun. Pure Appl. Math. 32 (1979) 797-838. | MR | Zbl

[23] J.J. Monaghan, Why particle methods work. SIAM J. Sci. Stat. Comput 3 (1982) 422-433. | MR | Zbl

[24] J.J. Monaghan, Extrapolating B-splines for interpolation. J. Comput. Phys. 60 (1985) 253-262. | MR | Zbl

[25] J.J. Monaghan, Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys. 30 (1992) 543-574. | Zbl

[26] P. Ploumhans, G.S. Winckelmans, J.K. Salmon, A. Leonard and M.S. Warren, Vortex methods for direct numerical simulation of three-dimensional bluff body flows : application to the sphere at Re = 300, 500, and 1000. J. Comput. Phys. 178 (2002) 427-463. | MR | Zbl

[27] P. Poncet, Topological aspects of the three-dimensional wake behind rotary oscillating circular cylinder. J. Fluid Mech. 517 (2004) 27-53. | MR | Zbl

[28] G.A. Sod, A survey of several finite-difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27 (1978) 1-131. | MR | Zbl

[29] L. Weynans, Méthode particulaire multi-niveaux pour la dynamique des gaz, application au calcul d'écoulements multifluides. Ph.D. thesis, Université Joseph Fourier. Available on : http://tel.archives-ouvertes.fr/tel-00121346/en/ (2006).

Cited by Sources: