Discretization correction of general integral PSE operators for particle methods. (English) Zbl 1334.65196

Summary: The general integral particle strength exchange (PSE) operators [J. D. Eldredge, A. Leonard and T. Colonius, J. Comput. Phys. 180, No. 2, 686–709 (2002; Zbl 1143.76550)] approximate derivatives on scattered particle locations to any desired order of accuracy. Convergence is, however, limited to a certain range of resolutions. For high-resolution discretizations, the constant discretization error dominates and prevents further convergence. We discuss a consistent discretization correction framework for PSE operators that yields the desired rate of convergence for any resolution, both on uniform Cartesian and irregular particle distributions, as well as near boundaries. These discretization-corrected (DC) PSE operators also have no overlap condition, enabling the kernel width to become arbitrarily small for constant interparticle spacing. We show that, on uniform Cartesian particle distributions, this leads to a seamless transition between DC PSE operators and classical finite difference stencils. We further identify relationships between DC PSE operators and operators used in corrected smoothed particle hydrodynamics and reproducing kernel particle methods. We analyze the presented DC PSE operators with respect to accuracy, rate of convergence, computational efficiency, numerical dispersion, numerical diffusion, and stability.


65N75 Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs
74S20 Finite difference methods applied to problems in solid mechanics
76M28 Particle methods and lattice-gas methods


Zbl 1143.76550


Matlab; LAPACK
Full Text: DOI Link


[1] Eldredge, J.D.; Leonard, A.; Colonius, T., A general deterministic treatment of derivatives in particle methods, J. comput. phys., 180, 686-709, (2002) · Zbl 1143.76550
[2] Degond, P.; Mas-Gallic, S., The weighted particle method for convection – diffusion equations. part 1: the case of an isotropic viscosity, Math. comput., 53, 188, 485-507, (1989) · Zbl 0676.65121
[3] Raviart, P.A., An analysis of particle methods, (), 244-323
[4] Cortez, R., Convergence of high-order deterministic particle methods for the convection – diffusion equation, Commun. pure appl. math. L, 1235-1260, (1997) · Zbl 0904.65140
[5] Cottet, G.H., A particle-grid superposition method for the navier – stokes equations, J. comput. phys., 89, 301-318, (1990) · Zbl 0699.76033
[6] Hieber, S.E.; Koumoutsakos, P., A Lagrangian particle level set method, J. comput. phys., 210, 342-367, (2005) · Zbl 1076.65087
[7] Bergdorf, M.; Cottet, G.-H.; Koumoutsakos, P., Multilevel adaptive particle methods for convection – diffusion equations, Multiscale model. simul., 4, 1, 328-357, (2005) · Zbl 1088.76055
[8] Degond, P.; Mas-Gallic, S., The weighted particle method for convection – diffusion equations. part 2: the anisotropic case, Math. comput., 53, 188, 509-525, (1989) · Zbl 0676.65122
[9] Sbalzarini, I.F.; Mezzacasa, A.; Helenius, A.; Koumoutsakos, P., Effects of organelle shape on fluorescence recovery after photobleaching, Biophys. J., 89, 3, 1482-1492, (2005)
[10] Poncet, P., Finite difference stencils based on particle strength schemes for improvement of vortex methods, J. turbul., 7, 23, 1-24, (2006) · Zbl 1273.76298
[11] Golia, C.; Buonomo, B.; Viviani, A., A corrected vortex blob method for 3D thermal buoyant flows, Energ. convers. manage., 49, 3243-3252, (2008)
[12] Koumoutsakos, P., Inviscid axisymmetrization of an elliptical vortex, J. comput. phys., 138, 821-857, (1997) · Zbl 0902.76080
[13] Wee, D.; Ghoniem, A.F., Modified interpolation kernels for treating diffusion and remeshing in vortex methods, J. comput. phys., 213, 239-263, (2006) · Zbl 1088.76050
[14] Johnson, G.R.; Beissel, S.R., Normalized smoothing functions for SPH impact computations, Int. J. numer. meth. eng., 39, 2725-2741, (1996) · Zbl 0880.73076
[15] Randles, P.W.; Libersky, L.D., Smoothed particle hydrodynamics: some recent improvements and applications, Comput. meth. appl. mech. eng., 139, 375-408, (1996) · Zbl 0896.73075
[16] Bonet, J.; Kulasegaram, S., Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations, Int. J. numer. meth. eng., 47, 1189-1214, (2000) · Zbl 0964.76071
[17] Oger, G.; Doring, M.; Alessandrini, B.; Ferrant, P., An improved SPH method: towards higher order convergence, J. comput. phys., 225, 1472-1492, (2007) · Zbl 1118.76050
[18] Lanson, N.; Vila, J.-P., Convergence des méthodes particulaires renormalisées pour LES systèmes de Friedrichs, C.R. acad. sci. Paris I, 340, 465-470, (2005) · Zbl 1068.65111
[19] Lanson, N.; Vila, J.-P., Renormalized meshfree schemes I: consistency, stability, and hybrid methods for conservation laws, SIAM J. numer. anal., 46, 4, 1912-1934, (2008) · Zbl 1178.65123
[20] Lanson, N.; Vila, J.-P., Renormalized meshfree schemes II: convergence for scalar conservation laws, SIAM J. numer. anal., 46, 4, 1935-1964, (2008) · Zbl 1178.65124
[21] Liu, W.K.; Jun, S.; Zhang, Y.F., Reproducing kernel particle methods, Int. J. numer. meth. fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[22] Liu, W.K.; Chen, Y.; Uras, R.A.; Chang, C.T., Generalized multiple scale reproducing kernel particle methods, Comput. meth. appl. mech. eng., 139, 91-157, (1996) · Zbl 0896.76069
[23] Gasser, T.; Müller, H.-G.; Mammitzsch, V., Kernels for nonparametric curve estimation, J.R. stat. soc. B, 47, 2, 238-252, (1985) · Zbl 0574.62042
[24] Mammitzsch, V., Optimal kernels, Stat. dec., 25, 153-172, (2007) · Zbl 1159.62020
[25] Fasshauer, G.E., Meshfree approximation methods with MATLAB, (2007), World Scientific · Zbl 1123.65001
[26] Demkowicz, L.; Karafiat, A.; Liszka, T., On some convergence results for FDM with irregular mesh, Comput. meth. appl. mech. eng., 42, 343-355, (1984) · Zbl 0518.65070
[27] Wright, G.B.; Fornberg, B., Scattered node compact finite difference-type formulas generated from radial basis functions, J. comput. phys., 212, 99-123, (2006) · Zbl 1089.65020
[28] T.-P. Fries, H.-G. Matthies, Classification and overview of meshfree methods, Informatikbericht 2003-03, Technical University Braunschweig (2004).
[29] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. meth. appl. mech. engrg., 139, 3-47, (1996) · Zbl 0891.73075
[30] Boyd, J.P., Chebyshev and Fourier spectral methods, (2001), Courier Dover Publications · Zbl 0987.65122
[31] Shankar, S.; van Dommelen, L., A new diffusion procedure for vortex methods, J. comput. phys., 127, 88-109, (1996) · Zbl 0859.76054
[32] Hou, T.Y., Convergence of a variable blob vortex method for the Euler and navier – stokes equations, SIAM J. numer. anal., 27, 6, 1387-1404, (1990) · Zbl 0727.76034
[33] Cottet, G.-H.; Koumoutsakos, P.; Ould Salihi, M.L., Vortex methods with spatially varying cores, J. comput. phys., 162, 164-185, (2000) · Zbl 1006.76070
[34] Verlet, L., Computer experiments on classical fluids. I. thermodynamical properties of lennard-Jones molecules, Phys. rev., 159, 1, 98-103, (1967)
[35] Marvasti, F.A., Nonuniform sampling – theory and practice, (2001), Springer · Zbl 0987.00028
[36] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, third ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999.
[37] Monaghan, J.J., Extrapolating B splines for interpolation, J. comput. phys., 60, 253-262, (1985) · Zbl 0588.41005
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