Structure and structure-preserving algorithms for plasma physics

TL;DR

This paper reviews Hamiltonian and action principle formulations in plasma physics, emphasizing structure-preserving numerical algorithms like symplectic, variational, and metriplectic integrators, with applications to plasma simulations such as GEMPIC.

Contribution

It provides a comprehensive overview of structure-preserving algorithms in plasma physics, including new insights into metriplectic and double bracket methods for equilibrium and dissipative systems.

Findings

Conservative and symplectic integrators exactly preserve invariants and geometric structures.

Metriplectic and double bracket dynamics effectively handle dissipative and equilibrium states.

Application of structure-preserving algorithms to plasma simulations enhances accuracy and stability.

Abstract

Hamiltonian and action principle (HAP) formulations of plasma physics are reviewed for the purpose of explaining structure preserving numerical algorithms. Geometric structures associated with and emergent from HAP formulations are discussed. These include conservative integration, which exactly conserves invariants, symplectic integration, which exactly preserves the Hamiltonian geometric structure, and other Hamiltonian integration techniques. Basic ideas of variational integration and Poisson integration, which can preserve noncanonical Hamiltonian structure, are discussed. Metriplectic integration, which preserves the structure of conservative systems with both Hamiltonian and dissipative parts, is proposed. Two kinds of simulated annealing, a relaxation technique for obtaining equilibrium states, are reviewed: one that uses metriplectic dynamics, which maximizes an entropy at fixed energy, the other that uses double bracket dynamics, which preserves Casimir invariants. Throughout, applications to plasma systems are emphasized. The paper culminates with a discussion of GEMPIC [Kraus et al., arXiv:1609.03053v1 [math.NA] (2016)], a particle in cell code that incorporates Hamiltonian and geometrical structure preserving properties.