On the kernel and particle consistency in smoothed particle hydrodynamics

Sigalotti, Leonardo Di G.; Klapp, Jaime; Rendón, Otto; Vargas, Carlos A.; Peña-Polo, Franklin

doi:10.1016/j.apnum.2016.05.007

Cited by 27 publications

(22 citation statements)

References 27 publications

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“…In particular, satisfaction of relation (5) means that the homogeneity of space is not affected by the SPH kernel approximation, which has as a consequence the conservation of linear momentum. On the other hand, fulfillment of relation (11) expresses that the isotropy of space is preserved by the kernel approximation, and therefore angular momentum is locally conserved (Vignjevic & Campbell 2009;Sigalotti et al 2016). …”

Section: The Issue Of Consistencymentioning

confidence: 99%

“…Making the change of variable -¢  -¢ r r r r h | | | |, it is easy to show that the following scaling relation holds (Sigalotti et al 2016):…”

Section: The Issue Of Consistencymentioning

confidence: 99%

“…In particular, restoring C 0 particle consistency implies that the homogeneity and isotropy of the discrete space are preserved, which has as a consequence the conservation of linear and angular momentum in practical calculations (Vignjevic & Campbell 2009;Sigalotti et al 2016). The goal here is to track the quality of the particle consistency relations in a true hydrodynamic evolution involving large density and pressure gradients as well as large spatial and temporal variations of the smoothing length.…”

Section: The Issue Of Consistencymentioning

confidence: 99%

“…Using Equation (1), it is a simple matter to show that the integral on the right of Equation (13), which corresponds to the second moment of the kernel M 2 , is equal to á ñ -á ñá ñ ¹ rr r r 0 2 ( ) , implying that C 2 consistency is not achieved even though C 0 and C 1 consistencies are automatically satisfied (Sigalotti et al 2016). This term is just the variance of the particle position vector r and is a measure of the spread of the particle positions relative to the mean.…”

Section: The Issue Of Consistencymentioning

confidence: 99%

“…Combining this with the leading error (∝h 2 ) of the continuous kernel approximation for most commonly used kernel forms, Zhu et al (2015) derived the scaling relations µ n N 1 2 and µ -h N 1 6 , which satisfy the joint limit as the domain resolution is progressively increased. A recent analysis on standard SPH has demonstrated that by using the above scalings C 0 consistency is fully restored for both the estimates of the function and its derivatives, in contrast to the case where n is fixed to a constant small value, with the numerical solution also becoming insensitive to the degree of particle disorder (Sigalotti et al 2016).…”

Section: Introductionmentioning

confidence: 99%

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Consistent SPH Simulations of Protostellar Collapse and Fragmentation

Gabbasov

Sigalotti

Cruz

et al. 2017

ApJ

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We study the consistency and convergence of smoothed particle hydrodynamics (SPH) as a function of the interpolation parameters, namely the number of particles N, the number of neighbors n, and the smoothing length h, using simulations of the collapse and fragmentation of protostellar rotating cores. The calculations are made using a modified version of the GADGET-2 code that employs an improved scheme for the artificial viscosity and power-law dependences of n and h on N, as was recently proposed by Zhu et al., which comply with the combined limit  ¥ N ,  h 0, and  ¥ n with  n N 0 for full SPH consistency as the domain resolution is increased. We apply this realization to the "standard isothermal test case" in the variant calculated by Burkert & Bodenheimer and the Gaussian cloud model of Boss to investigate the response of the method to adaptive smoothing lengths in the presence of large density and pressure gradients. The degree of consistency is measured by tracking how well the estimates of the consistency integral relations reproduce their continuous counterparts. In particular, C 0 and C 1 particle consistency is demonstrated, meaning that the calculations are close to second-order accuracy. As long as n is increased with N, mass resolution also improves as the minimum resolvable mass~-M n min 1 . This aspect allows proper calculation of small-scale structures in the flow associated with the formation and instability of protostellar disks around the growing fragments, which are seen to develop a spiral structure and fragment into close binary/multiple systems as supported by recent observations.

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Section: The Issue Of Consistencymentioning

confidence: 99%

“…Making the change of variable -¢  -¢ r r r r h | | | |, it is easy to show that the following scaling relation holds (Sigalotti et al 2016):…”

Section: The Issue Of Consistencymentioning

confidence: 99%

Section: The Issue Of Consistencymentioning

confidence: 99%

Section: The Issue Of Consistencymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Consistent SPH Simulations of Protostellar Collapse and Fragmentation

Gabbasov

Sigalotti

Cruz

et al. 2017

ApJ

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A diffusion‐equation based kernel function for smoothed particle hydrodynamics: Methods and validations

Cao

Gui

Zhang

et al. 2023

Numerical Methods in Fluids

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A diffusion‐equation‐based kernel function (called the diffusive kernel) is proposed for flow and heat transfer simulation by the smoothed‐particle‐hydrodynamics (SPH) method. This new diffusive function has basic features of physics‐based, intrinsically conservative, and mathematically smoothing, because of a general solution of the diffusion equation. Three cases have been employed for method validation. One is the SOD's problem (a one‐dimensional case of the Riemann problem). The SPH simulation with the new kernel is compared to both analytical solutions and the SPH simulation by the third‐order B‐spline kernel. The second one is the dam‐break case to compare the performance of the cubic‐spline kernel, the quintic‐spline kernel, and the Wendland C2 kernel with the current diffusive kernel. The present kernel seems to perform as well as the well‐known Wendland C2 and the quintic kernels. The last case is natural convection in a concentric circular domain with temperature differences between the inner and outer concentric walls. Both the Gaussian kernel and the diffusive kernel were used for comparison and validation. Good consistency between numerical and analytical results, as well as experimental results, validates the good performance of the current diffusive kernel. After validation, it is clear that the diffusive kernel can simulate both the shock wave and natural convective heat transfer very well, and is much better than the conventional Gaussian kernel except for computational efficiency. The kernel profiles, together with their derivatives and integrals, are explored to explain why the diffusive kernel's performance is much better. Finally, to increase numerical efficiency, a fitted expression of the diffusive kernel is also given. The fitted diffusive kernel can reach the same level of computational efficiency as the Gaussian kernel while maintaining the basic advantages of the current diffusive kernel, which gives a choice for practical application.

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