Integer Lattice Dynamics for Vlasov-Poisson

TL;DR

This paper revisits the integer lattice method for solving Vlasov-Poisson equations, demonstrating its efficiency, simplicity, and potential for high-dimensional collisionless system simulations.

Contribution

It introduces a variant of the IL method that is easy to implement, conservative, and reversible, with improvements to reduce numerical diffusion and memory scaling.

Findings

IL method is Lagrangian, conservative, and reversible.

Proposed N^4 scaling technique reduces memory requirements.

IL complements existing methods like N-body and finite volume schemes.

Abstract

We revisit the integer lattice (IL) method to numerically solve the Vlasov-Poisson equations, and show that a slight variant of the method is a very easy, viable, and efficient numerical approach to study the dynamics of self-gravitating, collisionless systems. The distribution function lives in a discretized lattice phase-space, and each time-step in the simulation corresponds to a simple permutation of the lattice sites. Hence, the method is Lagrangian, conservative, and fully time-reversible. IL complements other existing methods, such as N-body/particle mesh (computationally efficient, but affected by Monte-Carlo sampling noise and two-body relaxation) and finite volume (FV) direct integration schemes (expensive, accurate but diffusive). We also present improvements to the FV scheme, using a moving mesh approach inspired by IL, to reduce numerical diffusion and the time-step criterion. Being a direct integration scheme like FV, IL is memory limited (memory requirement for a full 3D problem scales as N^6, where N is the resolution per linear phase-space dimension). However, we describe a new technique for achieving N^4 scaling. The method offers promise for investigating the full 6D phase-space of collisionless systems of stars and dark matter.