A New Approach to Compute Linear Landau Damping

TL;DR

The paper presents a semi-analytical, pole-free method to compute the exact time-domain response of the linearized Vlasov–Maxwell system, focusing on ion dynamics with quasi-neutrality and adiabatic electrons. By solving the frequency-domain ion response via the plasma dispersion function $Z(\zeta)$ and employing a time-symmetric Fourier approach, it eliminates Gibbs artefacts and yields a numerically stable framework for both the ion density response and the full ion distribution-function evolution. The authors introduce a regularized principal-value formulation, add analytic pole contributions, and leverage time-reversal symmetry to construct solutions for arbitrary initial conditions, enabling high-precision benchmarks for six-dimensional kinetic codes. They demonstrate the approach with ion-sound and Langmuir-like scenarios and validate it against a semi-Lagrangian Vlasov code, highlighting improved numerical efficiency and reliability for rigorous code verification and calibration in plasma physics.

Abstract

Fully kinetic simulations of the Vlasov equation require a careful numerical treatment of phase space advections to ensure accuracy and stability in six dimensions. To test the accuracy of full Vlasov codes, we have developed a surprisingly simple, semi-analytical method for calculating the exact solution of the linearized Vlasov-Maxwell system in the time domain. In this work, we introduce the method by calculating the ion density response and the ion distribution function response to an initial ion density perturbation in an electrostatic setup without a magnetic field.

Figures (15)

• Figure 1: Illustration of the classical Landau damping test Landau:1946. The gradient of the straight blue line represents the damping according to the Landau pole. The red line indicates the time evolution of the absolute value of an initial pure electron density perturbation at a fixed position in the presence of a static ion background. After an initial transient period, the exponential decay is dominated by the Landau pole.
• Figure 2: The Dawson function along the real line. For all values $\zeta \in \mathbb{C}$, $D(\zeta)$ is entire, antisymmetric and asymptotically proportional to $1/\zeta$.
• Figure 3: Complex plot of $2 + \zeta Z(\zeta)$. Roots are indicated by points around which closed paths pass once through all complex phases. The complex phases are colour-coded using a rainbow spectrum.
• Figure 4: Fourier spectrum of the density response $n_1^+(\omega)$.
• Figure 5: Plot of the Fourier inversion of the unsymmetrized density response $n_1^+$ with numerical integration domain $[-20,20]$. At $t=0$, the solution converges to exactly half of the correct value.