From particle methods to forward-backward Lagrangian schemes

Campos Pinto, Martin; Charles, Frédérique

doi:10.5802/smai-jcm.31

Campos Pinto, Martin ¹ ; Charles, Frédérique ¹

¹ Sorbonne Université, Université Paris-Diderot SPC, CNRS, Laboratoire Jacques-Louis Lions, LJLL, F-75005 Paris

The SMAI Journal of computational mathematics, Tome 4 (2018), pp. 121-150.

Résumé

In this article we study a novel method for improving the accuracy of density reconstructions based on markers pushed forward by some available particle code. The method relies on the backward Lagrangian representation of the transported density, and it evaluates the backward flow using the current position of point particles seen as flow markers. Compared to existing smooth particle methods with either fixed or transformed shapes, the proposed reconstruction achieves higher locality and accuracy. This is confirmed by our error analysis which shows a theoretical gain of one convergence order compared to the LTP/QTP methods introduced in [8], and by numerical experiments that demonstrate significant CPU gains and an improved robustness relative to the remapping period.

Publié le : 2018-03-28

MR Zbl

DOI : 10.5802/smai-jcm.31

Affiliations des auteurs :

Campos Pinto, Martin ¹ ; Charles, Frédérique ¹

¹ Sorbonne Université, Université Paris-Diderot SPC, CNRS, Laboratoire Jacques-Louis Lions, LJLL, F-75005 Paris

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     author = {Campos Pinto, Martin and Charles, Fr\'ed\'erique},
     title = {From particle methods to forward-backward {Lagrangian} schemes},
     journal = {The SMAI Journal of computational mathematics},
     pages = {121--150},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {4},
     year = {2018},
     doi = {10.5802/smai-jcm.31},
     mrnumber = {3794744},
     zbl = {1416.65392},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/smai-jcm.31/}
}

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Campos Pinto, Martin; Charles, Frédérique. From particle methods to forward-backward Lagrangian schemes. The SMAI Journal of computational mathematics, Tome 4 (2018), pp. 121-150. doi : 10.5802/smai-jcm.31. http://www.numdam.org/articles/10.5802/smai-jcm.31/

Bibliographie
Cité par

[1] Alard, C.; Colombi, S. A cloudy Vlasov solution, Monthly Notices of the Royal Astronomical Society, Volume 359 (2005) no. 1, pp. 123-163 | DOI

[2] Bateson, W.B.; Hewett, D.W. Grid and Particle Hydrodynamics, Journal of Computational Physics, Volume 144 (1998), pp. 358-378 | DOI

[3] Beale, J.T.; Majda, A. Vortex methods. II. Higher order accuracy in two and three dimensions, Mathematics of Computation, Volume 39 (1982) no. 159, pp. 29-52 | MR

[4] Bergdorf, M.; Cottet, G.-H.; Koumoutsakos, P. Multilevel Adaptive Particle Methods for Convection-Diffusion Equations, Multiscale Modeling & Simulation, Volume 4 (2005) no. 1, pp. 328-357 | DOI | MR | Zbl

[5] Bergdorf, M.; Koumoutsakos, P. A Lagrangian particle-wavelet method, Multiscale Modeling & Simulation, Volume 5 (2006) no. 3, pp. 980-995 | DOI | MR | Zbl

[6] Biancalani, A.; Bottino, A.; Briguglio, S.; Koenies, A.; Lauber, Ph.; Mishchenko, A.; Poli, E.; Scott, B.D.; Zonca, F. Linear gyrokinetic particle-in-cell simulations of Alfven instabilities in tokamaks (2015) ( $〈$ arXiv:1510.01945 $〉$ )

[7] Bokanowski, O.; Garcke, J.; Griebel, M.; Klompmaker, I. An adaptive sparse grid semi-Lagrangian scheme for first order Hamilton-Jacobi Bellman equations, Journal of Scientific Computing, Volume 55 (2013) no. 3, pp. 575-605 | DOI | MR | Zbl

[8] Campos Pinto, M. Towards smooth particle methods without smoothing, Journal of Scientific Computing (2014) | MR | Zbl

[9] Campos Pinto, M.; Charles, F. Uniform Convergence of a Linearly Transformed Particle Method for the Vlasov–Poisson System, SIAM Journal on Numerical Analysis, Volume 54 (2016) no. 1, pp. 137-160 | MR | Zbl

[10] Campos Pinto, M.; Sonnendrücker, E.; Friedman, A.; Grote, D.P.; Lund, S.M. Noiseless Vlasov–Poisson simulations with linearly transformed particles, Journal of Computational Physics, Volume 275 (2014) no. C, pp. 236-256 | DOI | MR | Zbl

[11] Chen, Y.; Parker, S.E.; Rewoldt, G.; Ku, S.-H.; Park, G.-Y.; Chang, C.-S. Coarse-graining the electron distribution in turbulence simulations of tokamak plasmas, Physics of Plasmas, Volume 15 (2008) no. 5, 055905 pages | DOI

[12] Cohen, A.; Perthame, B. Optimal Approximations of Transport Equations by Particle and Pseudoparticle Methods, SIAM Journal on Mathematical Analysis, Volume 32 (2000) no. 3, pp. 616-636 | DOI | MR | Zbl

[13] Colombi, S.; Alard, C. A “metric” semi-Lagrangian Vlasov-Poisson solver (2016), pp. 1-28 (Submitted)

[14] Cotter, C.J.; Frank, J.; Reich, S. The remapped particle-mesh semi-Lagrangian advection scheme, Quarterly Journal of the Royal Meteorological Society, Volume 133 (2007) no. 622, pp. 251-260 | DOI

[15] Cottet, G.-H.; Koumoutsakos, P.; Salihi, M.L.O. Vortex Methods with Spatially Varying Cores, Journal of Computational Physics, Volume 162 (2000) no. 1, pp. 164-185 | DOI | Zbl

[16] Cottet, G.H.; Koumoutsakos, P. Vortex Methods: Theory and Practice, Cambridge University Press, Cambridge, 2000

[17] Crouseilles, N.; Respaud, T.; Sonnendrücker, E. A forward semi-Lagrangian method for the numerical solution of the Vlasov equation, Computer Physics Communications, Volume 180 (2009) no. 10, pp. 1730-1745 | DOI | MR | Zbl

[18] Denavit, J. Numerical Simulation of Plasmas with Periodic Smoothing in Phase Space, Journal of Computational Physics, Volume 9 (1972), pp. 75-98 | DOI | Zbl

[19] Filbet, F.; Sonnendrücker, E. Comparison of Eulerian Vlasov solvers, Computer Physics Communications, Volume 150 (2003), pp. 247-266 | DOI | MR | Zbl

[20] Gauger, Ch.; Leinen, P.; Yserentant, H. The finite mass method, SIAM J. Numer. Anal., Volume 37 (2000) no. 6, pp. 1768-1799 | DOI | MR | Zbl

[21] Hald, O.H. Convergence of Vortex Methods for Euler’s Equations. II, SIAM Journal on Numerical Analysis, Volume 16 (1979) no. 5, pp. 726-755 | DOI | MR

[22] Hockney, R.W.; Eastwood, J.W. Computer simulation using particles, Taylor & Francis, Inc, Bristol, PA, USA, 1988

[23] Hou, T.Y. Convergence of a Variable Blob Vortex Method for the Euler and Navier-Stokes Equations, SIAM Journal on Numerical Analysis, Volume 27 (1990) no. 6, pp. 1387-1404 | DOI | MR

[24] Koumoutsakos, P. Inviscid Axisymmetrization of an Elliptical Vortex, Journal of Computational Physics, Volume 138 (1997), pp. 821-857 | DOI | MR | Zbl

[25] Koumoutsakos, P. Inviscid axisymmetrization of an elliptical vortex, Journal of Computational Physics, Volume 138 (1997) no. 2, pp. 821-857 | DOI | MR | Zbl

[26] LeVeque, R.J. High-resolution conservative algorithms for advection in incompressible flow, SIAM Journal on Numerical Analysis (1996), pp. 627-665 | DOI | MR | Zbl

[27] Magni, A.; Cottet, G.-H. Accurate, non-oscillatory, remeshing schemes for particle methods, Journal of Computational Physics, Volume 231 (2012) no. 1, pp. 152-172 | DOI | MR | Zbl

[28] Monaghan, J.J. Extrapolating B. Splines for Interpolation, Journal of Computational Physics, Volume 60 (1985), 253 pages | DOI | MR | Zbl

[29] Nair, R.D.; Scroggs, J.S.; Semazzi, F.H.M. A forward-trajectory global semi-Lagrangian transport scheme, Journal of Computational Physics, Volume 190 (2003) no. 1, pp. 275-294 | DOI | MR

[30] Nevins, W.M.; Hammett, G.W.; Dimits, A.M.; Dorland, W.; Shumaker, D.E. Discrete particle noise in particle-in-cell simulations of plasma microturbulence, Physics of Plasmas, Volume 12 (2005), 122305 pages | DOI

[31] Qiu, J.-M.; Christlieb, A. A conservative high order semi-Lagrangian WENO method for the Vlasov equation, Journal of Computational Physics, Volume 229 (2010), pp. 1130-1149 | DOI | MR | Zbl

[32] Raviart, P.-A. An analysis of particle methods, Numerical methods in fluid dynamics (Como, 1983), Lecture Notes in Mathematics, Berlin, 1985, pp. 243-324 | DOI

[33] Selalib Semi-Lagrangian Library (http://selalib.gforge.inria.fr/)

[34] Sonnendrücker, E.; Roche, J.; Bertrand, P.; Ghizzo, A. The semi-Lagrangian method for the numerical resolution of the Vlasov equation, Journal of Computational Physics, Volume 149 (1999) no. 2, pp. 201-220 | DOI | MR | Zbl

[35] Staniforth, A.; Côté, J. Semi-Lagrangian integration schemes for atmospheric models-a review, Monthly weather review, Volume 119 (1991), pp. 2206-2223 | DOI

[36] Unser, M.; Daubechies, I. On the approximation power of convolution-based least squares versus interpolation, Signal Processing, IEEE Transactions on, Volume 45 (1997) no. 7, pp. 1697-1711 | DOI | Zbl

[37] Wang, B.; Miller, G.H.; Colella, P. A Particle-In-Cell method with adaptive phase-space remapping for kinetic plasmas, SIAM Journal on Scientific Computing, Volume 33 (2011), pp. 3509-3537 | DOI | MR | Zbl

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