Electrostatic PIC

In the electrostatic particle-in-cell method only the electric field is self-consistently updated with the particle motion. This approach uses the Poisson equation to obtain the electrostatic potential from the charge density (which is obtained directly from the simulation macro-particle). There are a few different variations of the electrostatic PIC method implemented in WarpX as outlined below. For details of the possible input parameters for each case see here.

Labframe

Poisson’s equation is solved in the lab frame with the charge density of all species combined. More specifically, the code solves:

\[\boldsymbol{\nabla}^2 \phi = - \rho/\epsilon_0 \qquad \boldsymbol{E} = - \boldsymbol{\nabla}\phi\]

Electromagnetostatic

Poisson’s equation is solved in the lab frame with the charge density of all species combined. Additionally the 3-component vector potential is solved in the Coulomb Gauge with the current density of all species combined to include self magnetic fields. More specifically, the code solves:

\[\begin{split}\boldsymbol{\nabla}^2 \phi = - \rho/\epsilon_0 \qquad \boldsymbol{E} = - \boldsymbol{\nabla}\phi \\ \boldsymbol{\nabla}^2 \boldsymbol{A} = - \mu_0 \boldsymbol{j} \qquad \boldsymbol{B} = \boldsymbol{\nabla}\times\boldsymbol{A}\end{split}\]

Effective Potential

Poisson’s equation is solved with a modified dielectric function (resulting in an “effective potential”) to create a semi-implicit scheme which is robust to the numerical instability seen in explicit electrostatic PIC when \(\Delta t \omega_{pe} > 2\). If this option is used the additional parameter warpx.effective_potential_factor can also be specified to set the value of \(C_{EP}\) (default 4). The method is stable for \(C_{EP} \geq 1\) regardless of \(\Delta t\), however, the larger \(C_{EP}\) is set, the lower the numerical plasma frequency will be and therefore care must be taken to not set it so high that the plasma mode hybridizes with other modes of interest. Details of the method can be found in Appendix A of Barnes [1] (note that in that paper the method is referred to as “semi-implicit electrostatic” but here it has been renamed to “effective potential” to avoid confusion with the semi-implicit method of Chen et al.). In short, the code solves:

\[\boldsymbol{\nabla}\cdot\left(1+\frac{C_{EP}}{4}\sum_{s \in \text{species}}(\omega_{ps}\Delta t)^2 \right)\boldsymbol{\nabla} \phi = - \rho/\epsilon_0 \qquad \boldsymbol{E} = - \boldsymbol{\nabla}\phi\]

Relativistic

Poisson’s equation is solved for each species in their respective rest frame. The corresponding field is mapped back to the simulation frame and will produce both E and B fields. More specifically, in the simulation frame, this is equivalent to solving for each species

\[\boldsymbol{\nabla}^2 - (\boldsymbol{\beta}\cdot\boldsymbol{\nabla})^2\phi = - \rho/\epsilon_0 \qquad \boldsymbol{E} = -\boldsymbol{\nabla}\phi + \boldsymbol{\beta}(\boldsymbol{\beta} \cdot \boldsymbol{\nabla}\phi) \qquad \boldsymbol{B} = -\frac{1}{c}\boldsymbol{\beta}\times\boldsymbol{\nabla}\phi\]

where \(\boldsymbol{\beta}\) is the average (normalized) velocity of the considered species (which can be relativistic). See, e.g., Vay [2] for more information.

[1]

D.C. Barnes. Improved c1 shape functions for simplex meshes. Journal of Computational Physics, 424:109852, 2021. URL: https://www.sciencedirect.com/science/article/pii/S0021999120306264, doi:https://doi.org/10.1016/j.jcp.2020.109852.

[2]

J.-L. Vay. Simulation Of Beams Or Plasmas Crossing At Relativistic Velocity. Physics of Plasmas, 15(5):56701, May 2008. doi:10.1063/1.2837054.