A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations

Carrie, Michael; Shadwick, B. A.

doi:10.1063/1.4938035

Title: A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations

Abstract

Here, we present a time-implicit numerical method to solve the relativistic Vlasov–Ampere system of equations on a two dimensional phase space grid. The time-splitting algorithm we use allows the generalization of the work presented here to higher dimensions keeping the linear aspect of the resulting discrete set of equations. The implicit method is benchmarked against linear theory results for the relativistic Landau damping for which analytical expressions using the Maxwell-Juttner distribution function are derived. We note that, independently from the shape of the distribution function, the relativistic treatment features collective behaviors that do not exist in the non relativistic case. The numerical study of the relativistic two-stream instability completes the set of benchmarking tests.

Authors:

Carrie, Michael ^[1]; Shadwick, B. A. ^[1]

Univ. of Nebraska-Lincoln, Lincoln, NE (United States)

Publication Date:: Mon Jan 04 00:00:00 EST 2016

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

Sponsoring Org.:: USDOE Office of Science (SC), Fusion Energy Sciences (FES); National Science Foundation (NSF)

OSTI Identifier:: 1334790

Alternate Identifier(s):: OSTI ID: 1234172

Grant/Contract Number:: SC0008382; FG02-08ER55000; PHY-1104683

Resource Type:: Accepted Manuscript

Journal Name:: Physics of Plasmas

Additional Journal Information:: Journal Volume: 23; Journal Issue: 1; Journal ID: ISSN 1070-664X

Publisher:: American Institute of Physics (AIP)

Country of Publication:: United States

Language:: English

Subject:: 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; relativistic Vlasov; implicit; Eulerian; electric fields; numerical solutions; Maxwell equations; particle distribution functions; cumulative distribution functions

Citation Formats


                    Carrie, Michael, and Shadwick, B. A. A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations.  United States: N. p., 2016. 
Web.  doi:10.1063/1.4938035.

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                    Carrie, Michael, & Shadwick, B. A. A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations.  United States.  https://doi.org/10.1063/1.4938035

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                    Carrie, Michael, and Shadwick, B. A. Mon .  
"A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations".  United States.  https://doi.org/10.1063/1.4938035.  https://www.osti.gov/servlets/purl/1334790.

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@article{osti_1334790,

  title        = {A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations},

  author       = {Carrie, Michael and Shadwick, B. A.},

  abstractNote = {Here, we present a time-implicit numerical method to solve the relativistic Vlasov–Ampere system of equations on a two dimensional phase space grid. The time-splitting algorithm we use allows the generalization of the work presented here to higher dimensions keeping the linear aspect of the resulting discrete set of equations. The implicit method is benchmarked against linear theory results for the relativistic Landau damping for which analytical expressions using the Maxwell-Juttner distribution function are derived. We note that, independently from the shape of the distribution function, the relativistic treatment features collective behaviors that do not exist in the non relativistic case. The numerical study of the relativistic two-stream instability completes the set of benchmarking tests.},

  doi          = {10.1063/1.4938035},

  journal      = {Physics of Plasmas},

  number       = 1,

  volume       = 23,

  place        = {United States},

  year         = {Mon Jan 04 00:00:00 EST 2016},

  month        = {Mon Jan 04 00:00:00 EST 2016}

}

Copy to clipboard

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Works referenced in this record:

Radiotherapy using a laser proton accelerator
conference, January 2008

Murakami, Masao; Hishikawa, Yoshio; Miyajima, Satoshi
LASER-DRIVEN RELATIVISTIC PLASMAS APPLIED FOR SCIENCE, INDUSTRY, AND MEDICINE: The 1st International Symposium, AIP Conference Proceedings
DOI: 10.1063/1.2958203

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

Cipiccia, Silvia; Islam, Mohammad R.; Ersfeld, Bernhard
Nature Physics, Vol. 7, Issue 11
DOI: 10.1038/nphys2090

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

Shoucri, M.
Communications in Nonlinear Science and Numerical Simulation, Vol. 13, Issue 1
DOI: 10.1016/j.cnsns.2007.04.004

Effects of the spatial grid in simulation plasmas
journal, October 1970

Langdon, A. Bruce
Journal of Computational Physics, Vol. 6, Issue 2
DOI: 10.1016/0021-9991(70)90024-0

Plasma Physics via Computer Simulation
book, January 1991

Birdsall, C. K.; Langdon, A. B.
DOI: 10.1887/0750301171

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

Cormier-Michel, Estelle; Shadwick, B. A.; Geddes, C. G. R.
Physical Review E, Vol. 78, Issue 1
DOI: 10.1103/PhysRevE.78.016404

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

The Initial Value Problem in Relativistic Plasma
journal, January 1975

Godfrey, B. B.; Newberger, B. S.; Taggart, K. A.
IEEE Transactions on Plasma Science, Vol. 3, Issue 4
DOI: 10.1109/TPS.1975.4316905

Wave-Breaking Amplitude of Relativistic Oscillations in a Thermal Plasma
journal, July 1988

Katsouleas, T.; Mori, W. B.
Physical Review Letters, Vol. 61, Issue 1
DOI: 10.1103/PhysRevLett.61.90

Analysis of transient and asymptotic behavior in relativistic Landau damping
journal, November 1995

Sartori, C.; Coppa, G. G. M.
Physics of Plasmas, Vol. 2, Issue 11
DOI: 10.1063/1.871028

The influence of finite temperature on strong relativistic plasma waves
journal, January 1998

Khachatryan, A. G.
Physics of Plasmas, Vol. 5, Issue 1
DOI: 10.1063/1.872679

A hybrid semi-Lagrangian Vlasov code. II. Numerical simulation of chirped beatwave experiment
journal, November 1998

Ghizzo, A.; Bertrand, P.; Lebas, J.
Physics of Plasmas, Vol. 5, Issue 11
DOI: 10.1063/1.873126

Study of propagation of ultraintense electromagnetic wave through plasma using semi-Lagrangian Vlasov codes
journal, August 2000

Huot, F.; Ghizzo, A.; Bertrand, P.
IEEE Transactions on Plasma Science, Vol. 28, Issue 4
DOI: 10.1109/27.893305

Steady ion momentum in nonlinear plasma waves
journal, February 2002

Gorbunov, L. M.; Mora, P.; Ramazashvili, R. R.
Physical Review E, Vol. 65, Issue 3
DOI: 10.1103/PhysRevE.65.036401

Theory of relativistic phase-space holes in a hot-electron-positron-ion plasma
journal, October 2005

Eliasson, B.; Shukla, P. K.
Physics of Plasmas, Vol. 12, Issue 10
DOI: 10.1063/1.2080607

Hamiltonian Description of Low-Temperature Relativistic Plasmas
journal, October 2004

Shadwick, B. A.; Tarkenton, G. M.; Esarey, E. H.
Physical Review Letters, Vol. 93, Issue 17
DOI: 10.1103/PhysRevLett.93.175002

Fluid and Vlasov models of low-temperature, collisionless, relativistic plasma interactions
journal, May 2005

Shadwick, B. A.; Tarkenton, G. M.; Esarey, E.
Physics of Plasmas, Vol. 12, Issue 5
DOI: 10.1063/1.1865032

Trapping, dark current, and wave breaking in nonlinear plasma waves
journal, March 2006

Schroeder, C. B.; Esarey, E.; Shadwick, B. A.
Physics of Plasmas, Vol. 13, Issue 3
DOI: 10.1063/1.2173960

Wave-breaking limits for relativistic electrostatic waves in a one-dimensional warm plasma
journal, December 2006

Trines, R. M. G. M.; Norreys, P. A.
Physics of Plasmas, Vol. 13, Issue 12
DOI: 10.1063/1.2398927

Theoretical and numerical studies of relativistic ion and electron holes in plasmas
journal, May 2007

Eliasson, B.; Shukla, P. K.
Physics of Plasmas, Vol. 14, Issue 5
DOI: 10.1063/1.2435989

Study on longitudinal dispersion relation in one-dimensional relativistic plasma: Linear theory and Vlasov simulation
journal, September 2013

Zhang, H.; Wu, S. Z.; Zhou, C. T.
Physics of Plasmas, Vol. 20, Issue 9
DOI: 10.1063/1.4821606

Principles of Plasma Physics
journal, December 1973

Krall, Nicholas A.; Trivelpiece, Alvin W.; Gross, Robert A.
American Journal of Physics, Vol. 41, Issue 12
DOI: 10.1119/1.1987587

Plasma oscillations
journal, January 1978

Case, K. M.
Physics of Fluids, Vol. 21, Issue 2
DOI: 10.1063/1.862199

On neutral plasma oscillations
journal, January 1994

Shadwick, B. A.; Morrison, P. J.
Physics Letters A, Vol. 184, Issue 3
DOI: 10.1016/0375-9601(94)90389-1

La nouvelle licence de géographie
journal, January 1946

Chabot, Georges
L'information géographique, Vol. 10, Issue 1
DOI: 10.3406/ingeo.1946.5138

Review of General education in a true society.
journal, July 1946

Jenkins, R. L.
American Journal of Orthopsychiatry, Vol. 16, Issue 3
DOI: 10.1111/j.1939-0025.1946.tb05416.x

Temperature stabilization of instabilities in force-free magnetospheres
journal, June 1977

Buschauer, R.; Benford, G.
Monthly Notices of the Royal Astronomical Society, Vol. 179, Issue 2
DOI: 10.1093/mnras/179.2.99

Linear wave dispersion laws in unmagnetized relativistic plasma: Analytical and numerical results
journal, May 2001

Bergman, Jan; Eliasson, Bengt
Physics of Plasmas, Vol. 8, Issue 5
DOI: 10.1063/1.1358313

Kinetic and collisional effects on the linear evolution of fast ignition relevant beam instabilities
journal, August 2008

Cottrill, L. A.; Langdon, A. B.; Lasinski, B. F.
Physics of Plasmas, Vol. 15, Issue 8
DOI: 10.1063/1.2953816

Towards a Relativistically Correct Characterization of Counterstreaming Plasmas. I. Distribution Functions
journal, May 2010

Lazar, M.; Stockem, A.; Schlickeiser, R.
The Open Plasma Physics Journal, Vol. 3, Issue 1
DOI: 10.2174/1876534301003010138

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

Cheng, C. Z.; Knorr, Georg
Journal of Computational Physics, Vol. 22, Issue 3
DOI: 10.1016/0021-9991(76)90053-X

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

Schumer, Joseph W.; Holloway, James Paul
Journal of Computational Physics, Vol. 144, Issue 2
DOI: 10.1006/jcph.1998.5925

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

Sircombe, N. J.; Arber, T. D.
Journal of Computational Physics, Vol. 228, Issue 13
DOI: 10.1016/j.jcp.2009.03.029

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

Strang, Gilbert
SIAM Journal on Numerical Analysis, Vol. 5, Issue 3
DOI: 10.1137/0705041

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

Kahan, William; Li, Ren-Cang
Mathematics of Computation, Vol. 66, Issue 219
DOI: 10.1090/S0025-5718-97-00873-9

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

Arber, T. D.; Vann, R. G. L.
Journal of Computational Physics, Vol. 180, Issue 1
DOI: 10.1006/jcph.2002.7098

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

Heath, R. E.; Gamba, I. M.; Morrison, P. J.
Journal of Computational Physics, Vol. 231, Issue 4
DOI: 10.1016/j.jcp.2011.09.020

A new conservative unsplit method for the solution of the Vlasov equation
journal, April 2006

Elkina, N. V.; Büchner, J.
Journal of Computational Physics, Vol. 213, Issue 2
DOI: 10.1016/j.jcp.2005.09.023

Works referencing / citing this record:

Transition from convective to absolute Raman instability via the longitudinal relativistic effect by using Vlasov-Maxwell simulations
journal, January 2018

Wang, Q.; Liu, Z. J.; Zheng, C. Y.
Physics of Plasmas, Vol. 25, Issue 1
DOI: 10.1063/1.5003130

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Title: A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations

Abstract

Citation Formats

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Gamma-rays from harmonically resonant betatron oscillations in a plasma wake
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Analysis of transient and asymptotic behavior in relativistic Landau damping
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The influence of finite temperature on strong relativistic plasma waves
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Theory of relativistic phase-space holes in a hot-electron-positron-ion plasma
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Hamiltonian Description of Low-Temperature Relativistic Plasmas
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Fluid and Vlasov models of low-temperature, collisionless, relativistic plasma interactions
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VALIS: A split-conservative scheme for the relativistic 2D Vlasov–Maxwell system
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