SPH Approximation of a Function

Let ${\displaystyle \prod f\left(x\right)}$ the integral approximation of a scalar function $f$ in space:

with $h$ the so-called smoothing length and $W$ a kernel approximation such that:

$\delta$ 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 ${\rho }_i$ . The smoothed approximation of the function $f$ is (summation over neighboring particles and the particle $i$ itself):

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

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

and

$$2h\le r\Rightarrow W\left(r,h\right)=0$$

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.