skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: 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 the 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, and the other that uses double bracket dynamics, which preserves Casimir invariants. Throughout, applications to plasma systems are emphasized. The paper concludes with a discussion of geometric electromagnetic particle-in-cell [Kraus et al., J. Plasma Phys. (to be published); e-print arXiv:1609.03053v1 [math.NA]], a particle in cell code that incorporates Hamiltonian and geometrical structure preserving properties.

Authors:
ORCiD logo [1]
  1. Univ. of Texas, Austin, TX (United States). Dept. of Physics and Inst. for Fusion Studies
Publication Date:
Research Org.:
Univ. of Texas, Austin, TX (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1535308
Grant/Contract Number:  
FG05-80ET53088
Resource Type:
Accepted Manuscript
Journal Name:
Physics of Plasmas
Additional Journal Information:
Journal Volume: 24; Journal Issue: 5; Journal ID: ISSN 1070-664X
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
physics

Citation Formats

Morrison, P. J. Structure and structure-preserving algorithms for plasma physics. United States: N. p., 2017. Web. doi:10.1063/1.4982054.
Morrison, P. J. Structure and structure-preserving algorithms for plasma physics. United States. doi:10.1063/1.4982054.
Morrison, P. J. Wed . "Structure and structure-preserving algorithms for plasma physics". United States. doi:10.1063/1.4982054. https://www.osti.gov/servlets/purl/1535308.
@article{osti_1535308,
title = {Structure and structure-preserving algorithms for plasma physics},
author = {Morrison, P. J.},
abstractNote = {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 the 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, and the other that uses double bracket dynamics, which preserves Casimir invariants. Throughout, applications to plasma systems are emphasized. The paper concludes with a discussion of geometric electromagnetic particle-in-cell [Kraus et al., J. Plasma Phys. (to be published); e-print arXiv:1609.03053v1 [math.NA]], a particle in cell code that incorporates Hamiltonian and geometrical structure preserving properties.},
doi = {10.1063/1.4982054},
journal = {Physics of Plasmas},
number = 5,
volume = 24,
place = {United States},
year = {2017},
month = {4}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 20 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Lie series and invariant functions for analytic symplectic maps
journal, December 1976

  • Dragt, Alex J.; Finn, John M.
  • Journal of Mathematical Physics, Vol. 17, Issue 12
  • DOI: 10.1063/1.522868

Hamiltonian–Dirac simulated annealing: Application to the calculation of vortex states
journal, January 2011


Action principles for relativistic extended magnetohydrodynamics: A unified theory of magnetofluid models
journal, February 2017

  • Kawazura, Yohei; Miloshevich, George; Morrison, Philip J.
  • Physics of Plasmas, Vol. 24, Issue 2
  • DOI: 10.1063/1.4975013

Numerical observation of turbulence enhanced growth rates
journal, July 1997

  • Doxas, Isidoros; Cary, John R.
  • Physics of Plasmas, Vol. 4, Issue 7
  • DOI: 10.1063/1.872230

A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth–Fokker–Planck equation
journal, September 2015


Lifting of the Vlasov–Maxwell bracket by Lie-transform method
journal, December 2016

  • Brizard, A. J.; Morrison, P. J.; Burby, J. W.
  • Journal of Plasma Physics, Vol. 82, Issue 6
  • DOI: 10.1017/S0022377816001161

What kinds of dynamics are there? Lie pseudogroups, dynamical systems and geometric integration
journal, October 2001


The Hamiltonian description of incompressible fluid ellipsoids
journal, August 2009

  • Morrison, P. J.; Lebovitz, Norman R.; Biello, Joseph A.
  • Annals of Physics, Vol. 324, Issue 8
  • DOI: 10.1016/j.aop.2009.04.003

Self-consistent chaos in the beam-plasma instability
journal, February 1994


Application of Newton's method to Lagrangian mappings
journal, August 1989


Comment on “Hamiltonian splitting for the Vlasov–Maxwell equations”
journal, September 2015


Hamiltonian particle-in-cell methods for Vlasov-Maxwell equations
journal, September 2016

  • He, Yang; Sun, Yajuan; Qin, Hong
  • Physics of Plasmas, Vol. 23, Issue 9
  • DOI: 10.1063/1.4962573

On the use of projectors for Hamiltonian systems and their relationship with Dirac brackets
journal, March 2013


Hamiltonian gyrokinetic Vlasov–Maxwell system
journal, September 2015


Generalized Hamiltonian Dynamics
journal, January 1950


Action principles for the Vlasov equation
journal, April 1992

  • Ye, Huanchun; Morrison, P. J.
  • Physics of Fluids B: Plasma Physics, Vol. 4, Issue 4
  • DOI: 10.1063/1.860231

Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov–Maxwell equations
journal, December 2015


Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media
journal, May 1966


Drift waves and transport
journal, April 1999


Hamiltonian and action principle formulations of plasma physics
journal, May 2005


Hamiltonian formulation of reduced magnetohydrodynamics
journal, January 1984

  • Morrison, P. J.; Hazeltine, R. D.
  • Physics of Fluids, Vol. 27, Issue 4
  • DOI: 10.1063/1.864718

Application of the phase space action principle to finite-size particle plasma simulations in the drift-kinetic approximation
journal, November 2014


Finite Element Hodge for spline discrete differential forms. Application to the Vlasov–Poisson system
journal, May 2014


A relativistic beam-plasma system with electromagnetic waves
journal, July 2005

  • Evstatiev, E. G.; Morrison, P. J.; Horton, W.
  • Physics of Plasmas, Vol. 12, Issue 7
  • DOI: 10.1063/1.1950127

Explicit high-order non-canonical symplectic particle-in-cell algorithms for Vlasov-Maxwell systems
journal, November 2015

  • Xiao, Jianyuan; Qin, Hong; Liu, Jian
  • Physics of Plasmas, Vol. 22, Issue 11
  • DOI: 10.1063/1.4935904

Symplectic integrators with adaptive time steps
journal, December 2011


Variational symplectic algorithm for guiding center dynamics and its application in tokamak geometry
journal, April 2009

  • Qin, Hong; Guan, Xiaoyin; Tang, William M.
  • Physics of Plasmas, Vol. 16, Issue 4
  • DOI: 10.1063/1.3099055

Quantum mechanics as a generalization of Nambu dynamics to the Weyl-Wigner formalism
journal, September 1991


Spectral Reduction: A Statistical Description of Turbulence
journal, December 1999


Variational principles of guiding centre motion
journal, February 1983


An introduction to Lie group integrators – basics, new developments and applications
journal, January 2014

  • Celledoni, Elena; Marthinsen, Håkon; Owren, Brynjulf
  • Journal of Computational Physics, Vol. 257
  • DOI: 10.1016/j.jcp.2012.12.031

Explicit Lie-Poisson integration and the Euler equations
journal, November 1993


Simulated annealing for three-dimensional low-beta reduced MHD equilibria in cylindrical geometry
journal, March 2017


Thoughts on brackets and dissipation: Old and new
journal, May 2009


Dissipative hamiltonian systems: A unifying principle
journal, February 1984


Hamiltonian description of the ideal fluid
journal, April 1998


Study of conservation and recurrence of Runge–Kutta discontinuous Galerkin schemes for Vlasov–Poisson systems
journal, January 2013

  • Cheng, Yingda; Gamba, Irene M.; Morrison, Philip J.
  • Journal of Scientific Computing, Vol. 56, Issue 2
  • DOI: 10.1007/s10915-012-9680-x

Fourth-order symplectic integration
journal, May 1990


Symplectic Runge-Kutta and related methods: recent results
journal, November 1992


Discrete mechanics and variational integrators
journal, May 2001


Entropy as a Metric Generator of Dissipation in Complete Metriplectic Systems
journal, August 2016


Exactly Conservative Integrators
journal, January 1998

  • Shadwick, B. A.; Bowman, John C.; Morrison, P. J.
  • SIAM Journal on Applied Mathematics, Vol. 59, Issue 3
  • DOI: 10.1137/S0036139995289313

A paradigm for joined Hamiltonian and dissipative systems
journal, January 1986


Geometric integration for particle accelerators
journal, April 2006


Weakly nonlinear dynamics in noncanonical Hamiltonian systems with applications to fluids and plasmas
journal, May 2016


Discontinuous Galerkin Methods for the Vlasov--Maxwell Equations
journal, January 2014

  • Cheng, Yingda; Gamba, Irene M.; Li, Fengyan
  • SIAM Journal on Numerical Analysis, Vol. 52, Issue 2
  • DOI: 10.1137/130915091

Geometric integration of the Vlasov-Maxwell system with a variational particle-in-cell scheme
journal, August 2012

  • Squire, J.; Qin, H.; Tang, W. M.
  • Physics of Plasmas, Vol. 19, Issue 8
  • DOI: 10.1063/1.4742985

Beatification: Flattening the Poisson bracket for two-dimensional fluid and plasma theories
journal, March 2017

  • Viscondi, Thiago F.; Caldas, Iberê L.; Morrison, Philip J.
  • Physics of Plasmas, Vol. 24, Issue 3
  • DOI: 10.1063/1.4977451

Integrators for Lie-Poisson dynamical systems
journal, May 1991


Energy-conserving numerical approximations for Vlasov plasmas
journal, August 1970


On the Hamiltonian formulation of incompressible ideal fluids and magnetohydrodynamics via Diracʼs theory of constraints
journal, January 2012


Numerical Integration of Lie--Poisson Systems While Preserving Coadjoint Orbits and Energy
journal, January 2001


Explicit high-order noncanonical symplectic algorithms for ideal two-fluid systems
journal, November 2016

  • Xiao, Jianyuan; Qin, Hong; Morrison, Philip J.
  • Physics of Plasmas, Vol. 23, Issue 11
  • DOI: 10.1063/1.4967276

Extremal energy properties and construction of stable solutions of the Euler equations
journal, October 1989


Hamiltonian time integrators for Vlasov-Maxwell equations
journal, December 2015

  • He, Yang; Qin, Hong; Sun, Yajuan
  • Physics of Plasmas, Vol. 22, Issue 12
  • DOI: 10.1063/1.4938034

Finite element exterior calculus: from Hodge theory to numerical stability
journal, January 2010

  • Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar
  • Bulletin of the American Mathematical Society, Vol. 47, Issue 2
  • DOI: 10.1090/S0273-0979-10-01278-4

Bracket formulation for irreversible classical fields
journal, February 1984


The energy‐momentum tensor for the linearized Maxwell–Vlasov and kinetic guiding center theories
journal, February 1991

  • Pfirsch, D.; Morrison, P. J.
  • Physics of Fluids B: Plasma Physics, Vol. 3, Issue 2
  • DOI: 10.1063/1.859735

The Heterognous Multiscale Methods
journal, January 2003


A symplectic integration algorithm for separable Hamiltonian functions
journal, January 1991


A general theory for gauge-free lifting
journal, January 2013


Symplectic integration of Hamiltonian systems
journal, May 1990


Errata and addenda: ``The local structure of Poisson manifolds''
journal, January 1985


Variational integration for ideal magnetohydrodynamics with built-in advection equations
journal, October 2014

  • Zhou, Yao; Qin, Hong; Burby, J. W.
  • Physics of Plasmas, Vol. 21, Issue 10
  • DOI: 10.1063/1.4897372

Nonlinear instability and chaos in plasma wave–wave interactions. I. Introduction
journal, June 1995

  • Kueny, C. S.; Morrison, P. J.
  • Physics of Plasmas, Vol. 2, Issue 6
  • DOI: 10.1063/1.871280

A guiding center Hamiltonian: A new approach
journal, December 1979

  • Littlejohn, Robert G.
  • Journal of Mathematical Physics, Vol. 20, Issue 12
  • DOI: 10.1063/1.524053

Hamiltonian formulation of the modified Hasegawa–Mima equation
journal, February 2014


The Hamiltonian structure of the Maxwell-Vlasov equations
journal, March 1982


Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics
journal, September 1980


Covariant poisson brackets for classical fields
journal, June 1986


Finite-dimensional collisionless kinetic theory
journal, March 2017


An exactly conservative integrator for the n -body problem
journal, September 2002


Collisionless Relaxation in Systems with Coulomb Interactions
journal, October 1970


Nonlinear instability and chaos in plasma wave–wave interactions. II. Numerical methods and results
journal, November 1995

  • Kueny, C. S.; Morrison, P. J.
  • Physics of Plasmas, Vol. 2, Issue 11
  • DOI: 10.1063/1.871039

Simulated annealing applied to two-dimensional low-beta reduced magnetohydrodynamics
journal, February 2015

  • Chikasue, Y.; Furukawa, M.
  • Physics of Plasmas, Vol. 22, Issue 2
  • DOI: 10.1063/1.4913234

Hamiltonian splitting for the Vlasov–Maxwell equations
journal, February 2015

  • Crouseilles, Nicolas; Einkemmer, Lukas; Faou, Erwan
  • Journal of Computational Physics, Vol. 283
  • DOI: 10.1016/j.jcp.2014.11.029

A method for Hamiltonian truncation: a four-wave example
journal, March 2016

  • Viscondi, Thiago F.; Caldas, Iberê L.; Morrison, Philip J.
  • Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 16
  • DOI: 10.1088/1751-8113/49/16/165501

A variational multi-symplectic particle-in-cell algorithm with smoothing functions for the Vlasov-Maxwell system
journal, October 2013

  • Xiao, Jianyuan; Liu, Jian; Qin, Hong
  • Physics of Plasmas, Vol. 20, Issue 10
  • DOI: 10.1063/1.4826218

The Maxwell-Vlasov equations as a continuous hamiltonian system
journal, December 1980


An Explicit Symplectic Integration Scheme for Plasma Simulations
journal, July 1993


Variational integrators for reduced magnetohydrodynamics
journal, September 2016


On the Stability of Plasma in Static Equilibrium
journal, January 1958

  • Kruskal, M. D.; Oberman, C. R.
  • Physics of Fluids, Vol. 1, Issue 4
  • DOI: 10.1063/1.1705885

Tokamak edge turbulence: background theory and computation
journal, June 2007


A Lagrangian theory for nonlinear wave packets in a collisionless plasma
journal, April 1972


A Can0nical Integrati0n Technique
journal, August 1983


Variational formulation of macro-particle plasma simulation algorithms
journal, May 2014

  • Shadwick, B. A.; Stamm, A. B.; Evstatiev, E. G.
  • Physics of Plasmas, Vol. 21, Issue 5
  • DOI: 10.1063/1.4874338

Metriplectic framework for dissipative magneto-hydrodynamics
journal, March 2012


Metriplectic structure, Leibniz dynamics and dissipative systems
journal, February 2007


Finite element exterior calculus, homological techniques, and applications
journal, May 2006


Symplectic maps, variational principles, and transport
journal, July 1992


Kinetic Guiding-Center Equations for the Theory of Drift Instabilities and Anomalous Transport
journal, February 1984


On Bogoliubov's kinetic equation for a spatially homogeneous plasma
journal, July 1960


Multiwave model for plasma–wave interaction
journal, October 2003

  • Evstatiev, E. G.; Horton, W.; Morrison, P. J.
  • Physics of Plasmas, Vol. 10, Issue 10
  • DOI: 10.1063/1.1609989

The local structure of Poisson manifolds
journal, January 1983


Construction of higher order symplectic integrators
journal, November 1990


Dynamics and thermodynamics of complex fluids.  I. Development of a general formalism
journal, December 1997


Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Examples with translation symmetry
journal, October 2016

  • Andreussi, T.; Morrison, P. J.; Pegoraro, F.
  • Physics of Plasmas, Vol. 23, Issue 10
  • DOI: 10.1063/1.4964900

Variational formulation of particle algorithms for kinetic plasma simulations
journal, July 2013


Local conservation laws for the Maxwell-Vlasov and collisionless kinetic guiding-center theories
journal, September 1985


Algebraic structure of the plasma quasilinear equations
journal, April 1982


Action principles for extended magnetohydrodynamic models
journal, September 2014

  • Keramidas Charidakos, I.; Lingam, M.; Morrison, P. J.
  • Physics of Plasmas, Vol. 21, Issue 9
  • DOI: 10.1063/1.4896336

Variational Formulation of Macroparticle Models for Electromagnetic Plasma Simulations
journal, June 2014

  • Stamm, Alexander B.; Shadwick, Bradley A.; Evstatiev, Evstati G.
  • IEEE Transactions on Plasma Science, Vol. 42, Issue 6
  • DOI: 10.1109/TPS.2014.2320461