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SPH Approximation of a Function

Let f(x) the integral approximation of a scalar function f in space:(1)
f(x)=Ωf(y)W(xy,h)dy
with h the so-called smoothing length and W a kernel approximation such that:(2)
x,ΩW(xy,h)dy=1
and in a suitable sense(3)
x,limh0W(xy,h)=δ(xy)

δ denotes the Dirac function.

Let a set of particles i =1, n at positions xi ( i =1,n) with mass mi and density ρi . The smoothed approximation of the function f is (summation over neighboring particles and the particle i itself):(4)
sf(x)=i=1,nmiρif(xi)W(xy,h)
The derivatives of the smoothed approximation are obtained by ordinary differentiation.(5)
f(x)=i=1,nmiρif(xi)W(xy,h)
The following kernel 1 which is an approximation of Gaussian kernel by cubic splines was chosen (Figure 1):(6)
rhW(r,h)=32πh3[23(rh)2+12(rh)3]
(7)
hr2hW(r,h)=14πh3(2rh)3
and (8)
2hrW(r,h)=0


Figure 1. Kernel Based on Spline Functions

This kernel has compact support, so that for each particle i , only the closest particles contribute to approximations at i (this feature is computationally efficient). The accuracy of approximating Equation 1 by Equation 4 depends on the order of the particles.

1
Monaghan J.J., “Smoothed Particle Hydrodynamics”, Annu.Rev.Astron.Astro-phys; Vol. 30; pp. 543-574, 1992.