Quasi-Neutrality Solver

The Quasi-Neutrality solver is designed to solve the following Quasi-Neutrality equation:

\[ -\Delta \phi(x) = \sum_s \int q_s f_s(x,y,v_x,v_y) dv_x dv_y \]

This calculation is split into two parts. Firstly the charge density is calculated:

\[ \rho_{q,s}(x, y) = \int q_s f_s(x,y,v_x,v_y) dv_x v_y \]

Then the basic Poisson equation:

\[ -\Delta \phi(x) = \sum_s \rho_{q,s}(x,y) \]

is solved.

Charge Density

The charge density is calculated by integrating the distribution function.

Quasi-Neutrality Solver

The Quasi-Neutrality equation can be solved with a variety of different methods. Here we have implemented:

FftQNSolver

These classes return the electric potential \(\phi\) and the electric field \(\frac{d \phi}{dx}\).

The FftQNSolver does not calculate the electric field using the Fourier modes. Rather it uses a spline interpolation to approximate this value. This interpolation is calculated by the operator ElectricField.