An unconditionally stable, time-implicit algorithm for solving the one-dimensional Vlasov–Poisson system

Carrié, M.; Shadwick, B. A.

doi:10.1017/S0022377821001124

Title: An unconditionally stable, time-implicit algorithm for solving the one-dimensional Vlasov–Poisson system

Abstract

The development of an implicit, unconditionally stable, numerical method for solving the Vlasov–Poisson system in one dimension using a phase-space grid is presented. The algorithm uses the Crank–Nicolson discretization scheme and operator splitting allowing for direct solution of the finite difference equations. This method exactly conserves particle number, enstrophy and momentum. A variant of the algorithm which does not use splitting also exactly conserves energy but requires the use of iterative solvers. This algorithm has no dissipation and thus fine-scale variations can lead to oscillations and the production of negative values of the distribution function. We find that overall, the effects of negative values of the distribution function are relatively benign. We consider a variety of test cases that have been used extensively in the literature where numerical results can be compared with analytical solutions or growth rates. We examine higher-order differencing and construct higher-order temporal updates using standard composition methods.

Authors:

Carrié, M.;

Publication Date:: Tue Mar 08 00:00:00 EST 2022

Research Org.:: Univ. of Nebraska, Lincoln, NE (United States)

Sponsoring Org.:: USDOE; National Science Foundation (NSF)

OSTI Identifier:: 1893545

Alternate Identifier(s):: OSTI ID: 1894688

Grant/Contract Number:: FG02-08ER55000; SC0008382

Resource Type:: Published Article

Journal Name:: Journal of Plasma Physics

Additional Journal Information:: Journal Name: Journal of Plasma Physics Journal Volume: 88 Journal Issue: 2; Journal ID: ISSN 0022-3778

Publisher:: Cambridge University Press (CUP)

Country of Publication:: United Kingdom

Language:: English

Subject:: 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; plasma simulation; plasma instabilities; plasma dynamics

Citation Formats


                    Carrié, M., and Shadwick, B. A. An unconditionally stable, time-implicit algorithm for solving the one-dimensional Vlasov–Poisson system.  United Kingdom: N. p., 2022. 
Web.  doi:10.1017/S0022377821001124.

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                    Carrié, M., & Shadwick, B. A. An unconditionally stable, time-implicit algorithm for solving the one-dimensional Vlasov–Poisson system.  United Kingdom.  https://doi.org/10.1017/S0022377821001124

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                    Carrié, M., and Shadwick, B. A. Tue .  
"An unconditionally stable, time-implicit algorithm for solving the one-dimensional Vlasov–Poisson system".  United Kingdom.  https://doi.org/10.1017/S0022377821001124.

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@article{osti_1893545,

  title        = {An unconditionally stable, time-implicit algorithm for solving the one-dimensional Vlasov–Poisson system},

  author       = {Carrié, M. and Shadwick, B. A.},

  abstractNote = {The development of an implicit, unconditionally stable, numerical method for solving the Vlasov–Poisson system in one dimension using a phase-space grid is presented. The algorithm uses the Crank–Nicolson discretization scheme and operator splitting allowing for direct solution of the finite difference equations. This method exactly conserves particle number, enstrophy and momentum. A variant of the algorithm which does not use splitting also exactly conserves energy but requires the use of iterative solvers. This algorithm has no dissipation and thus fine-scale variations can lead to oscillations and the production of negative values of the distribution function. We find that overall, the effects of negative values of the distribution function are relatively benign. We consider a variety of test cases that have been used extensively in the literature where numerical results can be compared with analytical solutions or growth rates. We examine higher-order differencing and construct higher-order temporal updates using standard composition methods.},

  doi          = {10.1017/S0022377821001124},

  journal      = {Journal of Plasma Physics},

  number       = 2,

  volume       = 88,

  place        = {United Kingdom},

  year         = {Tue Mar 08 00:00:00 EST 2022},

  month        = {Tue Mar 08 00:00:00 EST 2022}

}

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Title: An unconditionally stable, time-implicit algorithm for solving the one-dimensional Vlasov–Poisson system

Abstract

Citation Formats

Principles of Plasma Physics journal, December 1973

Conservative Numerical Schemes for the Vlasov Equation journal, September 2001

On the theory of stationary waves in plasmas journal, January 1955

On the velocity space discretization for the Vlasov–Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods journal, January 2016

Vlasov Simulations Using Velocity-Scaled Hermite Representations journal, August 1998

Iterative Refinement in Floating Point journal, April 1967

Thermal effects in plasma-based accelerators journal, May 2007

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form journal, November 2015

A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations journal, January 2016

Comparison of two Eulerian solvers for the four-dimensional Vlasov equation: Part I journal, February 2008

A finite element code for the simulation of one-dimensional vlasov plasmas. II. Applications journal, November 1988

Jacobian-free Newton–Krylov methods: a survey of approaches and applications journal, January 2004

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

Quadratic conservative scheme for relativistic Vlasov–Maxwell system journal, February 2019

Composition constants for raising the orders of unconventional schemes for ordinary differential equations journal, July 1997

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

Geometric Numerical Integration book, January 2002

Radiotherapy using a laser proton accelerator conference, January 2008

Charge-and-energy conserving moment-based accelerator for a multi-species Vlasov–Fokker–Planck–Ampère system, part I: Collisionless aspects journal, March 2015

Updating the Inverse of a Matrix journal, June 1989

Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations journal, June 1990

Gamma-rays from harmonically resonant betatron oscillations in a plasma wake journal, September 2011

A discontinuous Galerkin method for the Vlasov–Poisson system journal, February 2012

Eulerian codes for the numerical solution of the Vlasov equation journal, February 2008

Exactly Conservative Integrators journal, January 1998

VALIS: A split-conservative scheme for the relativistic 2D Vlasov–Maxwell system journal, July 2009

A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type journal, January 1947

Numerical solution of the Vlasov–Poisson system using generalized Hermite functions journal, October 2006

The free energy of Maxwell–Vlasov equilibria journal, June 1990

Physics of laser-driven plasma-based electron accelerators journal, August 2009

On the Construction and Comparison of Difference Schemes journal, September 1968

The Semi-Lagrangian Method for the Numerical Resolution of the Vlasov Equation journal, March 1999

Construction of higher order symplectic integrators journal, November 1990

Comparison of Eulerian Vlasov solvers journal, February 2003

The integration of the vlasov equation in configuration space journal, November 1976

Electrostatic Instabilities of a Uniform Non-Maxwellian Plasma journal, January 1960

A Numerical Scheme for the Integration of the Vlasov–Maxwell System of Equations journal, July 2002

Computationally efficient methods for modelling laser wakefield acceleration in the blowout regime journal, June 2012

Plasma Physics via Computer Simulation book, January 1991

Comparison of two Eulerian solvers for the four-dimensional Vlasov equation: Part II journal, February 2008

On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods journal, January 1995

Bound on the Energy Available from a Plasma journal, January 1963

Unphysical kinetic effects in particle-in-cell modeling of laser wakefield accelerators journal, July 2008

A Critical Comparison of Eulerian-Grid-Based Vlasov Solvers journal, July 2002

A numerical solution to the Vlasov equation journal, February 1999

Principles of Plasma Physics
journal, December 1973

Conservative Numerical Schemes for the Vlasov Equation
journal, September 2001

On the theory of stationary waves in plasmas
journal, January 1955

On the velocity space discretization for the Vlasov–Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods
journal, January 2016

Vlasov Simulations Using Velocity-Scaled Hermite Representations
journal, August 1998

Iterative Refinement in Floating Point
journal, April 1967

Thermal effects in plasma-based accelerators
journal, May 2007

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form
journal, November 2015

A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations
journal, January 2016

Comparison of two Eulerian solvers for the four-dimensional Vlasov equation: Part I
journal, February 2008

A finite element code for the simulation of one-dimensional vlasov plasmas. II. Applications
journal, November 1988

Jacobian-free Newton–Krylov methods: a survey of approaches and applications
journal, January 2004

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

Quadratic conservative scheme for relativistic Vlasov–Maxwell system
journal, February 2019

Composition constants for raising the orders of unconventional schemes for ordinary differential equations
journal, July 1997

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

Geometric Numerical Integration
book, January 2002

Radiotherapy using a laser proton accelerator
conference, January 2008

Charge-and-energy conserving moment-based accelerator for a multi-species Vlasov–Fokker–Planck–Ampère system, part I: Collisionless aspects
journal, March 2015

Updating the Inverse of a Matrix
journal, June 1989

Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations
journal, June 1990

Gamma-rays from harmonically resonant betatron oscillations in a plasma wake
journal, September 2011

A discontinuous Galerkin method for the Vlasov–Poisson system
journal, February 2012

Eulerian codes for the numerical solution of the Vlasov equation
journal, February 2008

Exactly Conservative Integrators
journal, January 1998

VALIS: A split-conservative scheme for the relativistic 2D Vlasov–Maxwell system
journal, July 2009

A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type
journal, January 1947

Numerical solution of the Vlasov–Poisson system using generalized Hermite functions
journal, October 2006

The free energy of Maxwell–Vlasov equilibria
journal, June 1990

Physics of laser-driven plasma-based electron accelerators
journal, August 2009

On the Construction and Comparison of Difference Schemes
journal, September 1968

The Semi-Lagrangian Method for the Numerical Resolution of the Vlasov Equation
journal, March 1999

Construction of higher order symplectic integrators
journal, November 1990

Comparison of Eulerian Vlasov solvers
journal, February 2003

The integration of the vlasov equation in configuration space
journal, November 1976

Electrostatic Instabilities of a Uniform Non-Maxwellian Plasma
journal, January 1960

A Numerical Scheme for the Integration of the Vlasov–Maxwell System of Equations
journal, July 2002

Computationally efficient methods for modelling laser wakefield acceleration in the blowout regime
journal, June 2012

Plasma Physics via Computer Simulation
book, January 1991

Comparison of two Eulerian solvers for the four-dimensional Vlasov equation: Part II
journal, February 2008

On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods
journal, January 1995

Bound on the Energy Available from a Plasma
journal, January 1963

Unphysical kinetic effects in particle-in-cell modeling of laser wakefield accelerators
journal, July 2008

A Critical Comparison of Eulerian-Grid-Based Vlasov Solvers
journal, July 2002

A numerical solution to the Vlasov equation
journal, February 1999