Structure and structure-preserving algorithms for plasma physics

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.

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Cited by in corpus (62)

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• Metriplectic Integrators for the Landau Collision Operator
• Symplectic integration with non-canonical quadrature for guiding-center orbits in magnetic confinement devices
• A General Metriplectic Framework with Application to Dissipative Extended Magnetohydrodynamics
• Continuum mechanics with torsion
• Quadratic conservative scheme for relativistic Vlasov--Maxwell system
• Fast neural Poincaré maps for toroidal magnetic fields
• Explicit Structure-Preserving Geometric Particle-in-Cell Algorithm in Curvilinear Orthogonal Coordinate Systems and Its Applications to Whole-Device 6D Kinetic Simulations of Tokamak Physics
• Exponential methods for solving hyperbolic problems with application to kinetic equations
• On Hamiltonian continuum mechanics
• Structure-preserving integrators for dissipative systems based on reversible-irreversible splitting
• Symplectic Gaussian Process Regression of Hamiltonian Flow Maps
• Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics
• Constrained Hamiltonian Systems and Physics-Informed Neural Networks: Hamilton-Dirac Neural Networks
• Field theory and structure-preserving geometric particle-in-cell algorithm for drift wave instability and turbulence
• Recovering Gardner Restacking with Purely Diffusive Operations
• Ehrenfest regularization of Hamiltonian systems
• An efficient, conservative, time-implicit solver for the fully kinetic arbitrary-species 1D-2V Vlasov-Ampère system
• Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria
• Hamiltonian Particle-in-Cell methods for Vlasov-Poisson equations
• Local Energy Conservation Law for Spatially-Discretized Hamiltonian Vlasov-Maxwell System
• Quasi-Helmholtz Decomposition, Gauss' Laws and Charge Conservation for Finite Element Particle-in-Cell
• Hamiltonian kinetic-Hall Magnetohydrodynamics with fluid and kinetic ions in the current and pressure coupling schemes
• Subcycling of particle orbits in variational, geometric electromagnetic particle-in-cell methods
• Advanced fuel fusion, phase space engineering, and structure-preserving geometric algorithms
• Eulerian variational formulations and momentum conservation laws for kinetic plasma systems
• Geometric Electrostatic Particle-In-Cell Algorithm on Unstructured Meshes
• Locally Structure-Preserving div-curl operators for high order Discontinuous Galerkin schemes
• Calculation of large-aspect-ratio tokamak and toroidally-averaged stellarator equilibria of high-beta reduced magnetohydrodynamics via simulated annealing
• Quasi-geometric integration of guiding-center orbits in piecewise linear toroidal fields
• Simulating nonlinear optical processes on a superconducting quantum device
• Conservative Projection Between Finite Element and Particle Bases
• Metriplectic foundations of gyrokinetic Vlasov-Maxwell-Landau theory
• Energy conserving particle-in-cell methods for relativistic Vlasov--Maxwell equations of laser-plasma interaction
• A class of three-dimensional gyroviscous magnetohydrodynamic models
• Poisson Structure of the Three-Dimensional Euler Equations in Fourier Space
• A gauge-compatible Hamiltonian splitting algorithm for particle-in-cell simulations using finite element exterior calculus
• Hamiltonian formulation of X-point collapse in an extended magnetohydrodynamics framework
• Gauge invariant canonical symplectic algorithms for real-time lattice strong-field quantum electrodynamics
• The Eulerian variational formulation of the gyrokinetic system in general spatial coordinates
• Variational Schemes and Geometric Simulations for a Hydrodynamic-Electrodynamic Model of Surface Plasmon Polaritons
• Exterior dissipation, proportional decay, and integrals of motion
• Towards a classification of bifurcations in Vlasov equations
• Variational Integration for Ideal Magnetohydrodynamics and Formation of Current Singularities
• Geometric Particle-in-Cell Simulations of the Vlasov--Maxwell System in Curvilinear Coordinates
• Linear stability analysis via simulated annealing and accelerated relaxation
• Variational formulation of classical and quantum models for intense laser pulse propagation
• Energy-momentum-conserving stochastic differential equations and algorithms for nonlinear Landau-Fokker-Planck equation
• Quantum computation of nonlinear maps
• Realization of Incompressible Navier-Stokes Flow as Superposition of Transport Processes for Clebsch Potentials
• Discrete Gravity with Local Lorentz Invariance
• Spatio-temporal Lie-Poisson discretization for incompressible magnetohydrodynamics on the sphere
• Charge-conserving, variational particle-in-cell method for the drift-kinetic Vlasov-Maxwell system
• Hamiltonian structure of the guiding-center Vlasov-Maxwell equations
• Symmetries and Local Conservation Laws of Variational Schemes for the Surface Plasmon Polaritons
• Symplectic particle-in-cell methods for hybrid plasma models with Boltzmann electrons and space-charge effects
• A Hamiltonian model for the macroscopic Maxwell equations using exterior calculus
• A stable semi-implicit algorithm
• Vlasov-Klimontovich Equation in Action
• Extended shallow-water theories with thermodynamics and geometry