Calculating the smoothing error in SPH

Abramowitz, 1972 Antuono, 2012, Numerical diffusive terms in weakly-compressible SPH schemes, Comput Phys Commun, 183, 2570, 10.1016/j.cpc.2012.07.006 Antuono, 2014, A measure of spatial disorder in particle methods, Comput Phys Commun, 185, 2609, 10.1016/j.cpc.2014.06.008 Benz, 1990, Smooth particle hydrodynamics: a review, in the numerical modeling of nonlinear stellar pulsation, NATO ASI Ser, 302, 269 Brookshaw, 1985, Solving the heat diffusion equation in SPH, Proc Astron Soc Aust, 6, 207, 10.1017/S1323358000018117 Cherfils, 2011 Colagrossi, 2009, Theoretical considerations on the free-surface role in the smoothed-particle-hydrodynamics model, Phys Rev E, 79, 056701, 10.1103/PhysRevE.79.056701 Colagrossi, 2011, Theoretical analysis and numerical verification of the consistency of viscous smoothed-particle-hydrodynamics formulations in simulating free-surface flows, Phys Rev E, 84, 086705 Colagrossi, 2017, Discussion of stokes hypothesis through the smoothed particle hydrodynamics model, Phys Rev E, 96, 023101, 10.1103/PhysRevE.96.023101 Cummins, 1999, An SPH projection method, J Comput Phys, 152, 584, 10.1006/jcph.1999.6246 Dehnen, 2012, Improving convergence in smoothed particle hydrodynamics simulations without pairing instability, Mon Not R Astron Soc, 425, 1068, 10.1111/j.1365-2966.2012.21439.x Español, 2003, Smoothed dissipative particle dynamics, Phys Rev E, 67, 10.1103/PhysRevE.67.026705 Franz, 2018, Convergence of the smoothed particle hydrodynamics method for a specific barotropic fluid flow: constructive kernel theory, SIAM J Math Anal, 50, 4752, 10.1137/17M1157696 Hashemi, 2016, Evaluation of a pressure splitting formulation for weakly compressible SPH: fluid flow around periodic array of cylinders, Comput Math with Appl, 71, 758, 10.1016/j.camwa.2015.12.034 Hu, 2006, A multiphase SPH method for macroscopic and mesoscopic flows, J Comput Phys, 213, 844, 10.1016/j.jcp.2005.09.001 Königsberger, 2004 Laskar, 2001, High order symplectic integrators for perturbed hamiltonian systems, Celest Mech Dyn Astron, 80, 39, 10.1023/A:1012098603882 Leroy, 2014, Unified semi-analytical wall boundary conditions applied to 2-d incompressible SPH, J Comput Phys, 261, 106, 10.1016/j.jcp.2013.12.035 Litvinov, 2015, Towards consistence and convergence of conservative SPH approximations, J Comput Phys, 301, 394, 10.1016/j.jcp.2015.08.041 Lubich, 1992, On convolution quadrature and Hille-Phillips operational calculus, Appl Numer Math, 9, 187, 10.1016/0168-9274(92)90014-5 Lubich, 2004, Convolution quadrature revisited, BIT Numer Math, 44, 503, 10.1023/B:BITN.0000046813.23911.2d Monaghan, 1985, Particle methods for hydrodynamics, Comput Phys Rep, 3, 71, 10.1016/0167-7977(85)90010-3 Monaghan, 1992, Smoothed particle hydrodynamics, Ann Rev Astron Astroph, 30, 543, 10.1146/annurev.aa.30.090192.002551 Monaghan, 2005, Smoothed particle hydrodynamics, Rep Prog Phys, 68, 1703, 10.1088/0034-4885/68/8/R01 Monaghan, 1983, Shock simulation by the particle method SPH, J Comput Phys, 52, 374, 10.1016/0021-9991(83)90036-0 Morris, 1997, Modelling low Reynolds number incompressible flows using SPH, J Comput Phys, 136, 214, 10.1006/jcph.1997.5776 Quinlan, 2006, Truncation error in mesh-free particle methods, Int J Numer Methods Eng, 66, 2064, 10.1002/nme.1617 Reed, 1975 Sun, 2019, A consistent approach to particle shifting in the δ-plus-SPH model, Comput Methods Appl Mech Eng, 348, 912, 10.1016/j.cma.2019.01.045 Violeau, 2009, Dissipative forces for Lagrangian models in computational fluid dynamics and application to smoothed-particle hydrodynamics, Phys Rev E, 80, 036705, 10.1103/PhysRevE.80.036705 Violeau, 2012 Violeau, 2014, On the maximum time step in weakly compressible SPH, J Comput Phys, 256, 388, 10.1016/j.jcp.2013.09.001 Violeau, 2015, Optimal time-step for incompressible SPH, J Comput Phys, 288, 119, 10.1016/j.jcp.2015.02.015 Violeau, 2015, SPH for free-surface flows: past, present and future, J Hydraul Res, 54, 1, 10.1080/00221686.2015.1119209 Violeau, 2017, Convergence rate of the SPH poisson equation on a cartesian grid Violeau, 2018, Spectral properties of the SPH laplacian operator, Comput Math Appl, 75, 3649, 10.1016/j.camwa.2018.02.023 Wendland, 1995, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv Comput Math, 4, 389, 10.1007/BF02123482