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, p. 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 : https://doi.org/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},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {1},
     year = {2013},
     pages = {57-81},
     doi = {10.1051/m2an/2012019},
     zbl = {1278.65136},
     mrnumber = {2968695},
     language = {en},
     url = {http://www.numdam.org/item/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/item/M2AN_2013__47_1_57_0/

[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 1740385 | Zbl 0949.65095

[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 2463201 | Zbl 1146.76038

[5] G.-H. Cottet and P.D. Koumoutsakos, Vortex methods. Cambridge University Press (2000). | MR 1755095 | Zbl 0953.76001

[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 2588783 | Zbl 1183.65102

[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 2241959 | Zbl 1096.65084

[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 1004.76066

[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 2203440 | Zbl 1088.76050

[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 0421.76032

[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 701178 | Zbl 0565.65050

[14] T. Hou and P.G. Lefloch, Why non-conservative schemes converge to wrong solutions : error analysis. Math. Comput. 62 (1994) 497-530. | MR 1201068 | Zbl 0809.65102

[15] P. Koumoutsakos and S. Hieber. A Lagrangian particle level set method. J. Comput. Phys. 210 (2005) 342-367. | MR 2157407 | Zbl 1076.65087

[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 0849.76061

[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 2135332 | Zbl 1068.65111

[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 2399402 | Zbl 1178.65124

[19] R.J. Leveque, Finite-volume methods for hyperbolic problems. Cambridge University Press (2002). | MR 1925043 | Zbl 1010.65040

[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 2846992 | Zbl pre06044227

[22] A. Majda and S. Osher, Numerical viscosity and the entropy condition. Commun. Pure Appl. Math. 32 (1979) 797-838. | MR 539160 | Zbl 0405.76021

[23] J.J. Monaghan, Why particle methods work. SIAM J. Sci. Stat. Comput 3 (1982) 422-433. | MR 677096 | Zbl 0498.76010

[24] J.J. Monaghan, Extrapolating B-splines for interpolation. J. Comput. Phys. 60 (1985) 253-262. | MR 805872 | Zbl 0588.41005

[25] J.J. Monaghan, Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys. 30 (1992) 543-574. | Zbl 0421.76032

[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 1899183 | Zbl 1045.76030

[27] P. Poncet, Topological aspects of the three-dimensional wake behind rotary oscillating circular cylinder. J. Fluid Mech. 517 (2004) 27-53. | MR 2260275 | Zbl 1131.76314

[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 495002 | Zbl 0387.76063

[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).