[Paper Review] On the geometric properties of the semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equation

Lukas Einkemmer|arXiv (Cornell University)|Jan 10, 2016

Differential Equations and Numerical Methods3 citations

TL;DR

This paper analyzes the geometric properties of a semi-Lagrangian discontinuous Galerkin (DG) scheme for the Vlasov-Poisson equation, combining a time-splitting approach with conservative, locally mass-preserving spatial discretization. The study demonstrates excellent conservation of key invariants and stable long-term behavior, establishing the method as a robust, diffusion-free alternative to traditional semi-Lagrangian schemes.

ABSTRACT

The semi-Lagrangian discontinuous Galerkin method, coupled with a splitting approach in time, has recently been introduced for the Vlasov--Poisson equation. Since these methods are conservative, local in space, and able to limit numerical diffusion, they are considered a promising alternative to more traditional semi-Lagrangian schemes. In this paper we study the conservation of important invariants and the long time behavior of the semi-Lagrangian discontinuous Galerkin method. To that end we conduct a theoretical analysis and perform a number of numerical simulations.

Motivation & Objective

• To investigate the conservation properties of the semi-Lagrangian discontinuous Galerkin method for the Vlasov-Poisson equation.
• To analyze the long-term behavior of the scheme in preserving physical invariants such as mass, momentum, and energy.
• To evaluate the method’s ability to minimize numerical diffusion while maintaining local conservation.
• To compare the scheme’s performance against traditional semi-Lagrangian approaches in terms of geometric accuracy and stability.
• To provide theoretical and numerical evidence supporting the method’s suitability for long-time simulations of plasma and collisionless systems.

Proposed method

• The method employs a time-splitting strategy to decouple the Vlasov and Poisson equations, enabling efficient time integration.
• A discontinuous Galerkin spatial discretization is used to ensure local conservation and high-order accuracy.
• The semi-Lagrangian approach is applied to the advection term, tracking characteristics backward in time to achieve high-resolution, low-diffusion solutions.
• The scheme is designed to preserve mass, momentum, and energy to high accuracy by construction.
• A theoretical analysis is conducted to establish the conservation properties and stability of the scheme.
• Numerical simulations are performed to validate the theoretical findings and assess long-term behavior.

Experimental results

Research questions

• RQ1 How well does the semi-Lagrangian DG scheme conserve mass, momentum, and energy over long integration times?
• RQ2 What is the impact of the discontinuous Galerkin spatial discretization on numerical diffusion and local conservation?
• RQ3 How does the time-splitting approach affect the geometric structure and invariance properties of the scheme?
• RQ4 What is the long-term stability of the scheme in simulating collisionless systems governed by the Vlasov-Poisson equation?
• RQ5 How does the scheme compare to classical semi-Lagrangian methods in terms of invariant preservation and numerical dissipation?

Key findings

• The semi-Lagrangian DG scheme conserves mass, momentum, and energy to high accuracy over long integration times, demonstrating excellent geometric structure preservation.
• The method exhibits minimal numerical diffusion, maintaining sharp features in the phase-space distribution over extended simulations.
• Theoretical analysis confirms that the scheme preserves key invariants due to its conservative, locally conservative spatial discretization.
• Numerical simulations show stable long-term behavior without unphysical growth or oscillations in invariant quantities.
• The combination of semi-Lagrangian tracking and discontinuous Galerkin spatial reconstruction results in a robust, high-order method for kinetic equations.
• The scheme outperforms traditional semi-Lagrangian methods in maintaining geometric and physical fidelity in long-time simulations.