Discontinuous Galerkin Representation of the Maxwell-Jüttner Distribution

TL;DR

Relativistic kinetic simulations require accurate representation of the Maxwell-Jüttner distribution on bounded momentum grids, but finite-domain truncation and projection errors can distort key moments. The authors introduce a moment-preserving iterative correction scheme compatible with discontinuous Galerkin discretizations, together with a robust, weak-projection-based method to compute nonlinear quantities such as the Lorentz boost factor $\Gamma$ and associated moments. They also develop a Maxwell-Jüttner initialization routine using asymptotic expansions to avoid problematic Bessel-function evaluations and a density-correction approach coupled with Picard iteration to force projected distributions to match target moments, achieving machine-precision accuracy with significantly reduced grid sizes. The resulting framework enables accurate, conservative relativistic BGK closures and Vlasov-Maxwell-BGK simulations on DG grids, with potential extensions to curved spaces and adaptive momentum-space grids.

Abstract

Kinetic simulations of relativistic gases and plasmas are critical for understanding diverse astrophysical and terrestrial systems, but the accurate construction of the relativistic Maxwellian, the Maxwell-Jüttner (MJ) distribution, on a discrete simulation grid is challenging. Difficulties arise from the finite velocity bounds of the domain, which may not capture the entire distribution function, as well as errors introduced by projecting the function onto a discrete grid. Here we present a novel scheme for iteratively correcting the moments of the projected distribution applicable to all grid-based discretizations of the relativistic kinetic equation. In addition, we describe how to compute the needed nonlinear quantities, such as Lorentz boost factors, in a discontinuous Galerkin (DG) scheme through a combination of numerical quadrature and weak operations. The resulting method accurately captures the distribution function and ensures that the moments match the desired values to machine precision.

Figures (4)

• Figure 1: Absolute error between the projected Maxwell-Jüttner distribution's moments and the desired moments, without the correction routine for higher moments. The moment values here are $n = 1.0, v_b = 0.5$, and $T = 1.0$ and the momentum bounds extend from $u_{max} = \pm 160$. At coarser resolutions, the initial non-monotonicity of the velocity error convergence is caused by small differences in the projection of the distribution onto the discrete grid.
• Figure 2: MJ plotted with the error between the corrected Maxwell-Jüttner distribution $f_c$ and the uncorrected distribution $f_{unc}$. Since the corrected, uncorrected, and theoretic distributions are indistinguishable on the plot, only the corrected distribution is plotted here.
• Figure 3: Example limitations on the MJ temperature supported by the finite momentum grids. The contours show the number of iterations required for the scheme to converge the moments to an absolute error of $\varepsilon < 10^{-12}$ at varied temperatures and grid parameters. White regions indicate the correction routine took greater than 20 iterations to converge or was unable to converge. Red lines overlay the temperature-limit estimates from this section. Panels (a), (c), and (e) show the non-relativistic limit, while panels (b), (d), and (f) consider relativistic distributions. The rows are ordered by increasing dimensionality of momentum space, from 1D, 2D and 3D. As a note for panel (e). The upper left corner, colored white, is simulations that were not run due to the large memory requirements. All panels were run with $n = 1$ and $\mathbf{v}_b = 0$.
• Figure 4: Reshaping of the distribution function from a water-bag to Maxwell-Jüttner due by the relativistic BGK operator. The plot includes three time slices: the initial state, one collision-time in, and ten collision-times into the simulation.