Smoothed-particle hydrodynamics

Definition

Let $W \colon \R^d \times (0, \infty) \to [0, \infty)$ be a smoothing kernel, that is, a smooth function $W(x, h)$ that is radial in $x$. It satisfies $\int_{\R^d} W(x, h) \, dx = 1$ for every $h > 0$, and it tends to the Dirac delta as $h \to 0$. Smoothed-particle hydrodynamics (SPH) approximates a fluid by $N$ particles with masses $m_a$, positions $x_a(t)$ and velocities $\mathbf{v}_a(t)$, for $a = 1, \dots, N$, as follows.

1. A field $A$ is approximated by $A(x) \approx \sum_b \frac{m_b}{\rho_b} A_b \, W(x - x_b, h)$, where $A_b$ and $\rho_b$ are the values at particle $b$.
2. The density at particle $a$ is $\rho_a = \sum_b m_b \, W(x_a - x_b, h)$.
3. The particles move with the flow, $\dot{x}_a = \mathbf{v}_a$, and, for the Euler equations of a barotropic fluid with pressure $p_a = P(\rho_a)$, their accelerations are $$\dot{\mathbf{v}}_a = -\sum_b m_b \left( \frac{p_a}{\rho_a^2} + \frac{p_b}{\rho_b^2} \right) \nabla_a W(x_a - x_b, h).$$