Electromagnetic PIC

In the electromagnetic Particle-In-Cell method [2, 3], the fields are updated using Maxwell’s equations:

(3)\[\frac{\partial \boldsymbol{B}}{\partial t} = -\nabla\times \boldsymbol{E}\]

(4)\[\frac{1}{c^2}\frac{\partial \boldsymbol{E}}{\partial t} = \nabla\times \boldsymbol{B}-\mu_0 \boldsymbol{j}\]

where \(\boldsymbol{E}\) and \(\boldsymbol{B}\) are the electric and magnetic field components, and \(\boldsymbol{j}\) is the current density.

Because the electromagnetic PIC method retains the full Maxwell equations, this method can capture the physics of the electromagnetic waves, including their propagation and self-consistent interaction with particles.

The electromagnetic PIC method can be run either with an explicit or implicit time integration scheme:

• In the explicit integration scheme, the particles and fields are updated sequentially at each time step (see Explicit electromagnetic PIC). This integration scheme is simple, but requires a small enough time step size \(\Delta t\) to ensure the stability of the simulation (e.g., CFL condition \(c\Delta t \lessapprox \Delta x\), need to resolve the plasma frequency \(\omega_p \Delta t \leq 2\) [2, 3]).

• In the implicit integration scheme, the particles and fields are updated simultaneously at each time step, using an iterative solver (see Implicit electromagnetic PIC). While this integration scheme is more complex, it can use larger time step sizes \(\Delta t\) and still retain the stability of the simulation. In addition, the implicit integration scheme is exactly energy conserving.

For more details, see the sections below:

• Explicit electromagnetic PIC
• Implicit electromagnetic PIC