The Linearized Vlasov-Maxwell System as a Hamiltonian System

TL;DR

The paper develops a structure-preserving, Hamiltonian formulation for the linearized Vlasov–Maxwell system with a Maxwellian background and curvilinear geometry. By combining a non-canonical Poisson bracket, FEEC-based field discretization, and PIC-like particle treatment within the GEMPIC framework, the authors achieve a semi-discrete system that conserves energy and respects the underlying geometric structure. Time integration via Poisson splitting yields an energy-stable sequence of substeps, and numerical experiments demonstrate accurate short-time damping, correct long-time energy behavior, and faithful reproduction of the Bernstein-wave spectrum, including curved geometries. The open-source STRUPHY implementation provides a practical tool for kinetic plasma simulations in realistic geometries with improved long-time stability and reduced noise compared to non-structure-preserving methods.

Abstract

We present a Hamiltonian formulation for the linearized Vlasov-Maxwell system with a Maxwellian background distribution function. We discuss the geometric properties of the model at the continuous level, and how to discretize the model in the GEMPIC framework [1]. This method allows us to preserve the structure of the system at the semi-discrete level. To integrate the model in time, we employ a Poisson splitting and discuss how to integrate each subsystem separately. We test the model against the direct delta-f method, which is the non-geometric pendant of our model. The first test case is the weak Landau damping, where our model exhibits the same physical properties for short simulations, but enjoys better long-time stability and energy conservation due to its geometric construction. These advantages becomes even more pronounced for the simulation of Bernstein waves, our second test case, where the noise in the direct delta-f method washes out all features of the dispersion relation whereas our model is able to reproduce the full spectrum correctly. The model is implemented in the open-source Python library STRUPHY [2], [3].

Figures (14)

• Figure 1: Sampled points from function \ref{['sampling-function']} without (left) and with (right) using of the constant function $f_0(x)=1$ as a control variate. Drawing with a control variate lowers the variance $\mathbb{V}$ with respect to the analytical function by a factor of $10^3$.
• Figure 2: Weak Landau damping of the electric field energy, including growth rates for the damping phase. Our model (right) reproduces the analytical damping rate of $m = - 0.3066$ and reaches the same damping level as the comparison model (left) until saturation. For comparison, the first (dominant) mode of the exact solution (black dashed line) is also shown.
• Figure 3: Long-time behaviour of the electric field energy in weak Landau damping. Our geometric model (right) keeps the amplitude of the energy at damped levels for long times, while the comparison model (left) shows an unphysical growth due to numerical noise after some time.
• Figure 4: Relative error of the total energy \ref{['relative-energy-error']} during weak Landau damping. Our model (right) keeps the error at the accuracy of the matrix inversion solver, while the comparison model (left) has a much larger error in total energy.
• Figure 5: The evolution of the distribution function in phase space for different times during weak Landau damping. The filamentation can be clearly seen.

Theorems & Definitions (2)

• Proposition 1
• Proposition 2