SPH Approximation of a Function

Let

\prod f (x)

the integral approximation of a scalar function

f

in space:(1)

\prod f (x) = \int_{Ω} f (y) W (x - y, h) d y

with

h

the so-called smoothing length and

W

a kernel approximation such that:(2)

\forall x, \int_{Ω} W (x - y, h) d y = 1

and in a suitable sense(3)

\forall x, \lim_{h \to 0} W (x - y, h) = δ (x - y)

$δ$ denotes the Dirac function.

Let a set of particles

i

=1, n at positions

x_{i}

(

i

=1,n) with mass

m_{i}

and density

ρ_{i}

. The smoothed approximation of the function

f

is (summation over neighboring particles and the particle

i

itself):(4)

\prod_{s} f (x) = \sum_{i = 1, n} \frac{m_{i}}{ρ_{i}} f (x_{i}) W (x - y, h)

The derivatives of the smoothed approximation are obtained by ordinary differentiation.(5)

\nabla f (x) = \sum_{i = 1, n} \frac{m_{i}}{ρ_{i}} f (x_{i}) \nabla W (x - y, h)

The following kernel ¹ which is an approximation of Gaussian kernel by cubic splines was chosen (Figure 1):(6)

r \leq h \Rightarrow W (r, h) = \frac{3}{2 π h^{3}} [\frac{2}{3} - {(\frac{r}{h})}^{2} + \frac{1}{2} {(\frac{r}{h})}^{3}]

(7)

h \leq r \leq 2 h \Rightarrow W (r, h) = \frac{1}{4 π h^{3}} {(2 - \frac{r}{h})}^{3}

and (8)

2 h \leq r \Rightarrow W (r, h) = 0

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.