parsys

This is an experimental particle system simulator. The API is very likely to change so using it in your project is currently not recommended.

Integration step

Two integration methods are implemented: Explicit Euler (EE) and Runge-Kutta 4 (RK4). EE is faster but less accurate.

Explicit Euler (EE)

For a simulation step from $t$ to $t + h$, we need as input:

• $\vec{x}^t$: Positions of all particles at time $t$.
• $\vec{v}^t$: Velocities of all particles at time $t$.
• $\vec{a}(\vec{x}, \vec{v})$: A function that, given all positions and velocities, computes the accelerations.

As output, we will compute approximations of $\vec{x}^{t + h}$ and $\vec{v}^{t + h}$.

Explicit Euler integration works as follows:

• Compute $\vec{a}^t = \vec{a}(\vec{x}^t, \vec{v}^t)$
• Approximate (discretization error $O(h^2)$):
  • $\vec{x}^{t + h} \approx \vec{x}^t + h \vec{v}^t$
  • $\vec{v}^{t + h} \approx \vec{v}^t + h \vec{a}^t$

Runge-Kutta (RK4)

For a simulation step from $t$ to $t + h$, we need as input:

• $\vec{x}^t$: Positions of all particles at time $t$.
• $\vec{v}^t$: Velocities of all particles at time $t$.
• $\vec{a}(\vec{x}, \vec{v})$: A function that, given all positions and velocities, computes the accelerations.

As output, we will compute approximations of $\vec{x}^{t + h}$ and $\vec{v}^{t + h}$.

RK4 integration works as follows:

• Compute $\vec{a}^t = \vec{a}(\vec{x}^t, \vec{v}^t)$
• Compute $\vec{x}^{mid_1} = \vec{x}^t + \frac{h}{2}\vec{v}^t$ and $\vec{v}^{mid_1} = \vec{v}^t + \frac{h}{2}\vec{a}^t$
• Compute $\vec{a}^{mid_1} = \vec{a}(\vec{x}^{mid_1}, \vec{v}^{mid_1})$
• Compute $\vec{x}^{mid_2} = \vec{x}^t + \frac{h}{2}\vec{v}^{mid_1}$ and $\vec{v}^{mid_2} = \vec{v}^t + \frac{h}{2}\vec{a}^{mid_1}$
• Compute $\vec{a}^{mid_2} = \vec{a}(\vec{x}^{mid_2}, \vec{v}^{mid_2})$
• Compute $\vec{x}^{end} = \vec{x}^t + h\vec{v}^{mid_2}$ and $\vec{v}^{end} = \vec{v}^t + h\vec{a}^{mid_2}$
• Compute $\vec{a}^{end} = \vec{a}(\vec{x}^{end}, \vec{v}^{end})$
• Approximate (discretization error $O(h^5)$):
  • $\vec{x}^{t + h} \approx \vec{x}^t + h \frac{\vec{v}^t + 2 \vec{v}^{mid_1} + 2 \vec{v}^{mid_2} + \vec{v}^{end}}{6}$
  • $\vec{v}^{t + h} \approx \vec{v}^t + h \frac{\vec{a}^t + 2 \vec{a}^{mid_1} + 2 \vec{a}^{mid_2} + \vec{a}^{end}}{6}$