A Particle-in-cell Method for Plasmas with a Generalized Momentum Formulation, Part II: Enforcing the Lorenz Gauge Condition

Abstract

In a previous paper, we developed a new particle-in-cell method for the Vlasov-Maxwell system in which the electromagnetic fields and the equations of motion for the particles were cast in terms of scalar and vector potentials through a Hamiltonian formulation. This paper extends this new class of methods by focusing on the enforcement the Lorenz gauge condition in both exact and approximate forms using co-located meshes. A time-consistency property of the proposed field solver for the vector potential form of Maxwell's equations is established, which is shown to preserve the equivalence between the semi-discrete Lorenz gauge condition and the analogous semi-discrete continuity equation. Using this property, we present three methods to enforce a semi-discrete gauge condition. The first method introduces an update for the continuity equation that is consistent with the discretization of the Lorenz gauge condition. The second approach we propose enforces a semi-discrete continuity equation using the boundary integral solution to the field equations. The third approach introduces a gauge correcting method that makes direct use of the gauge condition to modify the scalar potential and uses local maps for both the charge and current densities. The vector potential coming from the current density is taken to be exact, and using the Lorenz gauge, we compute a correction to the scalar potential that makes the two potentials satisfy the gauge condition. We demonstrate two of the proposed methods in the context of periodic domains. Problems defined on bounded domains, including those with complex geometric features remain an ongoing effort. However, this work shows that it is possible to design computationally efficient methods that can effectively enforce the Lorenz gauge condition in an non-staggered PIC formulation.

Figures (4)

• Figure 1: Growth in the magnetic field energy for the Weibel instability. We compare the growth rate in the $\ell_{2}$-norm of the magnetic field $B^{(3)}$ for different methods against an analytical growth rate predicted from linear response theory yoon1987exact. The analytical growth rate for this configuration is determined to be $\text{Im}(\omega) \approx 0.319734$. We observe good agreement with the theoretically predicted growth rate for each of the methods.
• Figure 2: The gauge error for different methods applied to the Weibel instability test problem. Naive interpolation introduces significant gauge error, whereas the maps based on the continuity equation or gauge correction result in substantial improvement. It should be noted that while FFT and FD6 appear identical, they differ by $\mathcal{O}(10^{-14})$. The gauge correcting approach produced the smallest gauge error among the methods we considered.
• Figure 3: A Maxwellian distribution of electrons and ions are placed on a periodic domain. The electrons are given a thermal velocity in addition to a drift velocity $\textbf{v}_{d} = (v_{d}^{(1)}, v_{d}^{(2)})^T$, where $v^{(1)} = v^{(2)} = c/100$, that is the speed is $\lVert \textbf{v}_{d} \rVert = \sqrt{2}c/100$. We see the particles escape the well only to fall back in later. The plot (b) at $t=0.05$ is their second such traversal. By $t=0.1$ the electrons have become quite dispersed throughout the domain, though the remnants of the original drift velocity can be found in a slight beam along the diagonal ($t=0.5)$.
• Figure 4: The gauge error of a cloud of electrons drifting into and out of a potential well. Naively interpolated such that the charge and current densities are not consistent, we see a significant gauge error. However, if the charge density is computed from the current density using the continuity equation and a high order method (eg FFT) to compute the divergence of $\textbf{J}$, we see a much better gauge error over time.

Theorems & Definitions (6)

• Lemma 2.1
• proof
• Theorem 2.1
• proof
• Theorem 2.2
• proof