Convergence of a particle-in-cell scheme for the spherically symmetric Vlasov-Einstein system

TL;DR

This work rigorously analyzes a particle-in-cell discretization of the spherically symmetric Vlasov–Einstein system, proving convergence of the semi-discrete scheme to the true solution and providing explicit error bounds that depend on the discretization parameters $\varepsilon$ and $\delta$. By recasting the Vlasov part via characteristic flows and expressing the Einstein source terms with smeared particle contributions, the authors derive an autonomous ODE system for particle-like variables $(R_n,W_n,M_n)$ and establish stability and consistency through a sequence of lemmas, including intermediate double-barred quantities to compare discretizations with the continuum. They then extend the analysis to a fully discrete time-stepping scheme, giving Euler-like updates and showing that discretization errors scale linearly with the time step $\tau$ under suitable assumptions. The results provide a rigorous mathematical foundation for the numerical approach used in prior simulations and offer concrete guidance on discretization parameters to achieve desired accuracy, contributing to the reliability of simulations of relativistic kinetic matter in general relativity.

Abstract

We consider spherically symmetric, asymptotically flat space-times with a collisionless gas as matter model. Many properties of the resulting Vlasov-Einstein system are not yet accessible by purely analytical means. We present a discretized version of this system which is suitable for numerical implementation and is based on the particle-in-cell technique. Convergence of the resulting approximate solutions to the exact solution is proven and error bounds are provided.

Theorems & Definitions (17)

• Lemma 2.1
• Theorem 2.2
• Lemma 3.1
• Theorem 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7
• Lemma 3.8