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Towards Smooth Particle Methods Without Smoothing

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Abstract

In this article we present a new class of particle methods which aim at being accurate in the uniform norm with a minimal amount of smoothing. The crux of our approach is to compute local polynomial expansions of the characteristic flow to transport the particle shapes with improved accuracy. In the first order case the method consists of representing the transported density with linearly-transformed particles, the second order version transports quadratically-transformed particles, and so on. For practical purposes we provide discrete versions of the resulting LTP and QTP schemes that only involve pointwise evaluations of the forward characteristic flow, and we propose local indicators for the associated transport error. On a theoretical level we extend these particle schemes up to arbitrary polynomial orders and show by a rigorous analysis that for smooth flows the resulting methods converge in \(L^\infty \) without requiring remappings, extended overlapping or vanishing moments for the particles. Numerical tests using different passive transport problems demonstrate the accuracy of the proposed methods compared to basic particle schemes, and they establish their robustness with respect to the remapping period. In particular, it is shown that QTP particles can be transported without remappings on very long periods of time, without hampering the accuracy of the numerical solutions. Finally, a dynamic criterion is proposed to automatically select the time steps where the particles should be remapped. The strategy is a by-product of our error analysis, and it is validated by numerical experiments.

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References

  1. Alard, C., Colombi, S.: A cloudy Vlasov solution. Mon. Not. R. Astron. Soc. 359(1), 123–163 (2005)

    Article  Google Scholar 

  2. Beale, J., Majda, A.: Vortex methods. II. Higher order accuracy in two and three dimensions. Math. Comput. 39(159), 29–52 (1982)

    MathSciNet  MATH  Google Scholar 

  3. Beale, J.T.: On the accuracy of vortex methods at large times. In: Computational Fluid Dynamics and Reacting Gas Flows (Minneapolis. MN, 1986), pp. 19–32. Springer, New York (1988)

  4. Bergdorf, M., Cottet, G.H., Koumoutsakos, P.: Multilevel adaptive particle methods for convection–diffusion equations. Multiscale Model. Simul. 4(1), 328–357 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bergdorf, M., Koumoutsakos, P.: A Lagrangian particle-wavelet method. Multiscale Model. Simul. 5(3), 980–995 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bokanowski, O., Garcke, J., Griebel, M., Klompmaker, I.: An adaptive sparse grid semi-Lagrangian scheme for first order Hamilton–Jacobi Bellman equations. J. Sci. Comput. 55(3), 575–605 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Campos Pinto, M., Charles, F.: Uniform convergence of a linearly transformed particle method for the Vlasov–Poisson system. hal-01080732 (2014)

  8. Campos Pinto, M., Sonnendrücker, E., Friedman, A., Grote, D.P., Lund, S.: Noiseless Vlasov–Poisson simulations with linearly transformed particles. J. Comput. Phys. 275(C), 236–256 (2014)

    Article  MathSciNet  Google Scholar 

  9. Chorin, A.: Numerical study of slightly viscous flow. J. Fluid Mech. 57(4), 785–796 (1973)

    Article  MathSciNet  Google Scholar 

  10. Chui, C., Diamond, H.: A characterization of multivariate quasi-interpolation formulas and its applications. Numer. Math. 57(1), 105–121 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cohen, A., Perthame, B.: Optimal approximations of transport equations by particle and pseudoparticle methods. SIAM J. Math. Anal. 32(3), 616–636 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cotter, C., Frank, J., Reich, S.: The remapped particle-mesh semi-Lagrangian advection scheme. Q. J. R. Meteorol. Soc. 133(622), 251–260 (2007)

    Article  Google Scholar 

  13. Cottet, G., Koumoutsakos, P.: Vortex Methods: Theory and Practice. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  14. Cottet, G.H.: Artificial viscosity models for vortex and particle methods. J. Comput. Phys. 127, 299–308 (1996)

  15. Cottet, G.H., Koumoutsakos, P., Salihi, M.: Vortex methods with spatially varying cores. J. Comput. Phys. 162(1), 164–185 (2000)

    Article  MATH  Google Scholar 

  16. Crouseilles, N., Respaud, T., Sonnendrücker, E.: A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Comput. Phys. Commun. 180(10), 1730–1745 (2009)

    Article  MATH  Google Scholar 

  17. Denavit, J.: Numerical simulation of plasmas with periodic smoothing in phase space. J. Comput. Phys. 9, 75–98 (1972)

    Article  MATH  Google Scholar 

  18. Hockney, R., Eastwood, J.: Computer Simulation Using Particles. Taylor & Francis, Bristol (1988)

    Book  MATH  Google Scholar 

  19. Holm, D., Nitsche, M., Putkaradze, V.: Euler-alpha and vortex blob regularization of vortex filament and vortex sheet motion. J. Fluid Mech. 555, 149–176 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hou, T.: Convergence of a variable blob vortex method for the Euler and Navier–Stokes equations. SIAM J. Numer. Anal. 27(6), 1387–1404 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Koumoutsakos, P.: Inviscid axisymmetrization of an elliptical vortex. J. Comput. Phys. 138, 821–857 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Langdon, A., Birdsall, C.: Plasma Physics Via Computer Simulation. Taylor & Francis, New York (2005)

    Google Scholar 

  23. LeVeque, R.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33, 627–665 (1996)

  24. Magni, A., Cottet, G.H.: Accurate, non-oscillatory, remeshing schemes for particle methods. J. Comput. Phys. 231(1), 152–172 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Monaghan, J.: Extrapolating B. Splines for interpolation. J. Comput. Phys. 60, 253 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nair, R., Scroggs, J., Semazzi, F.: A forward-trajectory global semi-Lagrangian transport scheme. J. Comput. Phys. 190(1), 275–294 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rasio, F.: Particle methods in astrophysical fluid dynamics. In: Progress of Theoretical Physics Supplement. MIT, Dept Phys, Cambridge, MA 02139 USA, pp. 609–621 (2000)

  28. Raviart, P.A.: An analysis of particle methods. In: Numerical Methods in Fluid Dynamics (Como, 1983), Lecture Notes in Mathematics, Berlin, pp. 243–324 (1985)

  29. Strain, J.: 2D vortex methods and singular quadrature rules. J. Comput. Phys. 124(1), 131–145 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Unser, M., Daubechies, I.: On the approximation power of convolution-based least squares versus interpolation. IEEE Trans. Signal Process. 45(7), 1697–1711 (1997)

    Article  MATH  Google Scholar 

  31. Van Rees, W., Leonard, A., Pullin, D., Koumoutsakos, P.: A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical flows at high Reynolds numbers. J. Comput. Phys. 230(8), 2794–2805 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, B., Miller, G., Colella, P.: A particle-in-cell method with adaptive phase-space remapping for kinetic plasmas. SIAM J. Sci. Comput. 33, 3509–3537 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author thanks Eric Sonnendrücker, Albert Cohen, Jean-Marie Mirebeau and Jean Roux for valuable discussions during the early stages of this research.

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, UMR 7598, CNRS, 75005, Paris, France

    Martin Campos Pinto

  2. Laboratoire Jacques-Louis Lions, UMR 7598, UPMC Univ Paris 06, Sorbonne Universités, 75005, Paris, France

    Martin Campos Pinto

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  1. Martin Campos Pinto
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Correspondence to Martin Campos Pinto.

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Campos Pinto, M. Towards Smooth Particle Methods Without Smoothing. J Sci Comput 65, 54–82 (2015). https://doi.org/10.1007/s10915-014-9953-7

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  • DOI: https://doi.org/10.1007/s10915-014-9953-7

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