A particle-in-Fourier method with semi-discrete energy conservation for non-periodic boundary conditions

TL;DR

A novel particle-in-Fourier (PIF) scheme that extends its applicability to non-periodic boundary conditions via standard potential theory is introduced and an order of magnitude speedup is shown compared to the standard PIC scheme for a long time integration cyclotron simulation.

Abstract

We introduce a novel particle-in-Fourier (PIF) scheme that extends its applicability to non-periodic boundary conditions. Our method handles free space boundary conditions by replacing the Fourier Laplacian operator in PIF with a mollified Green's function as first introduced by Vico-Greengard-Ferrando. This modification yields highly accurate free space solutions to the Vlasov-Poisson system, while still maintaining energy conservation up to an error bounded by the time step size. We also explain how to extend our scheme to arbitrary Dirichlet boundary conditions via standard potential theory, which we illustrate in detail for Dirichlet boundary conditions on a circular boundary. We support our approach with proof-of-concept numerical results from two-dimensional plasma test cases to demonstrate the accuracy, efficiency, and conservation properties of the scheme. By avoiding grid heating and finite grid instability we are able to show an order of magnitude speedup compared to the standard PIC scheme for a long time integration cyclotron simulation.

Figures (16)

• Figure 1: Our simulation domain based on the shape function and mollified Green's function chosen. Particles are within the domain $\Omega$ (green), and $S * S * \sum_{j}\delta(\textbf{x}-\textbf{X}_j)$ has a support $\Omega_{2R}$ (blue). The radius of the Green's function is chosen such that $g^L$ is able to cover the entire blue region $\Omega_{2R}$ whenever the center of $g^L$ is located inside the green region $\Omega$. To determine the smallest possible radius, we place the center of $g^L$ at the upper right corner of $\Omega$. After a simple calculation, we find the smallest radius to be $\sqrt{2}+2R$ for $g^L$. Finally, the extended domain is $\tilde{\Omega}$ (gray).
• Figure 2: Potential $\varphi$ from the analytic solution given by equation \ref{['eq:ana']} with particles represented as blue circles (top) and numerical error of our free space Poisson solver using different numbers of modes (bottom). We choose the radius of the Green's function to be $L=1.75$, and the solver uses a $(4N_m)^2$ box.
• Figure 3: Global errors in $L_2$ norms versus the number of modes in each dimension $N_m$, calculated in an upscaled physical grid of size $512\times 512$ using Fourier interpolation. Notice that without performing the precomputation step, the solver achieves spectral accuracy, whereas with the precomputation step, it has second order convergence.
• Figure 4: Charge density $\rho$ at different times of an infinitely long non-neutral beam confined by a strong constant and uniform magnetic field, simulated using 40,000 particles and $N_m=N_g=128$ using free space PIC and PIF methods. For the FSPIF scheme, the precomputation step is not performed.
• Figure 5: Absolute values of Fourier modes when $N_m=128$ for the same nonneutral beam simulation as in Figure \ref{['fig:fig5']}. These figures indicate that most high-frequency modes are not very significant, and there is no obvious mode-coupling or aliasing effects.