Topic/Type: 1. Plasma Simulation, Poster

### Grid-Free Electromagnetic Particle Simulations

A.J. Christlieb, B.W. Ong

Michigan State University, Department of Mathematics - USA

Basic plasma science plays an increasingly significant role in applications of importance to the United States Air Force. Many of these applications require a fully kinetic description in at least part of the domain. The most common approach is to use a fully Lagrangian framework, where the model is reduced to tracking the evolution of test particles in phase space. Of the many varieties, the most accepted approach is Particle-In-Cell. The PI, with his collaborators, is developing an alternative approach, the Boundary Integral Treecode (BIT). BIT is based on fast summation algorithms and boasts arbitrary accuracy.

In this talk, we demonstrate how the existing BIT framework can be extended to electrodynamic problems. The key idea is to consider the transpose of the standard method of lines methodology, i.e., we choose to discretize in time and directly solve the resulting Helmholtz equation using an integral formulation. Observe that Maxwell\'s equations can be cast as
$\small \frac{1}{c^2} E_{tt} - \nabla^2 E = - \mu_o J_t - \frac{1}{\epsilon_o}\nabla \rho$ ,
$\small \frac{1}{c^2} B_{tt} - \nabla^2 B =\mu_o \nabla \times J$,
where $\small \rho=\int f\,dv$ is the charge density, $\small J=\int v f\, dv$ is the current density, ($\small \epsilon_o$, $\small \mu_o$) are the permittivity and permeability and $\small c$ is the speed of light. To illustrate the implicit treecode time stepping methodology for $\small E$ and $\small B$, it suffices to consider the wave equation, $\small u_{tt}-k^2 \nabla^2 u =0$. Using a centered difference approximation to $\small u_{tt}$ and evaluating $\small \nabla^2 u$ at time level $\small n+1$ gives,
$\small \nabla^2 u^{n+1}-\frac{1}{k^2 \Delta t^2} u^{n+1}= \frac{1}{k^2 \Delta t^2 } \left ( -2 u^{n} +u^{n-1} \right )$.
The integral solution for $\small u^{n+1}$ is
$\small u^{n+1}(x) = \iint_{\Omega} \left( \frac{-2u^n+u^{n-1}}{k^2 \Delta t^2} \right ) G(x|y)\, d\Omega_y + \oint_{tial \Omega} (u^{n+1} \nabla G - G \nabla u^{n+1})\cdot {\bf n} \, ds,$
where $\small G(x|y)$, the free space Green\'s function for the Helmholtz operator $\small \mathcal{L}(\cdot) = (\nabla^2 - \frac{1}{k^2\Delta t^2})(\cdot)$, is $\small G(x|y)=\gamma \exp (- r/ (k \Delta t) ) /r$ in $\small \mathbb{R}^3$. Here, $\small r=||x-y||_2$ and $\small \gamma$ is the normalization. The volumetric and boundary integrals are approximated at the midpoints, and the resulting sums can be computed using fast summation algorithms which are $\small O(N \log N)$ time in evaluating the sum. We have applied this approach to the wave equation in 1D, 2D and 3D with Direcltet boundary conditions. For the wave equation, the method is able to take time steps much larger than the imposed CFL restrictions for an explicit integrator. Additionally, this implicit time stepping methodology has been extended to higher order approximations in time. We working on extned these ideas to Maxwell's equations.