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Title: Higher-order time integration of Coulomb collisions in a plasma using Langevin equations

Abstract

The extension of Langevin-equation Monte-Carlo algorithms for Coulomb collisions from the conventional Euler-Maruyama time integration to the next higher order of accuracy, the Milstein scheme, has been developed, implemented, and tested. This extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the two fixed-frame spherical-coordinate velocity variables. Results from the numerical implementation show the expected improvement [O(Δt) vs. O(Δt1/2)] in the strong convergence rate both for the speed |v| and angular components of the scattering. An important result is that this improved convergence is achieved for the angular component of the scattering if and only if the “area-integral” terms in the Milstein scheme are included. The resulting Milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency. These include both algorithms with improved convergence in the averages (weak convergence) and multi-time-level schemes. The latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional Monte-Carlo schemes, and their performance is improved considerably when the Milstein algorithm is used for the underlying time advance versus the Euler-Maruyama algorithm. A new method for sampling the area integrals is given which ismore » a simplification of an earlier direct method and which retains high accuracy. Lastly, this method, while being useful in its own right because of its relative simplicity, is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration.« less

Authors:
 [1];  [1];  [2];  [2];  [2]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
  2. Univ. of California, Los Angeles, CA (United States). Dept. of Mathematics
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1227022
Report Number(s):
LLNL-JRNL-577312
Journal ID: ISSN 0021-9991
Grant/Contract Number:  
AC52-07NA27344; FG02-05ER25710
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 242; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; 70 PLASMA PHYSICS AND FUSION; Monte-Carlo methods; Milstein method; Collision processes; Plasmas; Collisions; Computer applications

Citation Formats

Dimits, A. M., Cohen, B. I., Caflisch, R. E., Rosin, M. S., and Ricketson, L. F.. Higher-order time integration of Coulomb collisions in a plasma using Langevin equations. United States: N. p., 2013. Web. doi:10.1016/j.jcp.2013.01.038.
Dimits, A. M., Cohen, B. I., Caflisch, R. E., Rosin, M. S., & Ricketson, L. F.. Higher-order time integration of Coulomb collisions in a plasma using Langevin equations. United States. https://doi.org/10.1016/j.jcp.2013.01.038
Dimits, A. M., Cohen, B. I., Caflisch, R. E., Rosin, M. S., and Ricketson, L. F.. Fri . "Higher-order time integration of Coulomb collisions in a plasma using Langevin equations". United States. https://doi.org/10.1016/j.jcp.2013.01.038. https://www.osti.gov/servlets/purl/1227022.
@article{osti_1227022,
title = {Higher-order time integration of Coulomb collisions in a plasma using Langevin equations},
author = {Dimits, A. M. and Cohen, B. I. and Caflisch, R. E. and Rosin, M. S. and Ricketson, L. F.},
abstractNote = {The extension of Langevin-equation Monte-Carlo algorithms for Coulomb collisions from the conventional Euler-Maruyama time integration to the next higher order of accuracy, the Milstein scheme, has been developed, implemented, and tested. This extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the two fixed-frame spherical-coordinate velocity variables. Results from the numerical implementation show the expected improvement [O(Δt) vs. O(Δt1/2)] in the strong convergence rate both for the speed |v| and angular components of the scattering. An important result is that this improved convergence is achieved for the angular component of the scattering if and only if the “area-integral” terms in the Milstein scheme are included. The resulting Milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency. These include both algorithms with improved convergence in the averages (weak convergence) and multi-time-level schemes. The latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional Monte-Carlo schemes, and their performance is improved considerably when the Milstein algorithm is used for the underlying time advance versus the Euler-Maruyama algorithm. A new method for sampling the area integrals is given which is a simplification of an earlier direct method and which retains high accuracy. Lastly, this method, while being useful in its own right because of its relative simplicity, is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration.},
doi = {10.1016/j.jcp.2013.01.038},
journal = {Journal of Computational Physics},
number = C,
volume = 242,
place = {United States},
year = {Fri Feb 08 00:00:00 EST 2013},
month = {Fri Feb 08 00:00:00 EST 2013}
}

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Works referenced in this record:

Theory of plasma transport in toroidal confinement systems
journal, April 1976


The physics basis for ignition using indirect-drive targets on the National Ignition Facility
journal, February 2004

  • Lindl, John D.; Amendt, Peter; Berger, Richard L.
  • Physics of Plasmas, Vol. 11, Issue 2
  • DOI: 10.1063/1.1578638

Collisional losses of ring current ions
journal, January 1996

  • Jordanova, V. K.; Kistler, L. M.; Kozyra, J. U.
  • Journal of Geophysical Research: Space Physics, Vol. 101, Issue A1
  • DOI: 10.1029/95JA02000

Langevin Representation of Coulomb Collisions in PIC Simulations
journal, December 1997

  • Manheimer, Wallace M.; Lampe, Martin; Joyce, Glenn
  • Journal of Computational Physics, Vol. 138, Issue 2
  • DOI: 10.1006/jcph.1997.5834

A Monte-Carlo method for coulomb collisions in hybrid plasma models
journal, February 2008


Small-angle Coulomb collision model for particle-in-cell simulations
journal, March 2009

  • Lemons, Don S.; Winske, Dan; Daughton, William
  • Journal of Computational Physics, Vol. 228, Issue 5
  • DOI: 10.1016/j.jcp.2008.10.025

Time-Step Considerations in Particle Simulation Algorithms for Coulomb Collisions in Plasmas
journal, September 2010

  • Cohen, Bruce I.; Dimits, Andris M.; Friedman, Alex
  • IEEE Transactions on Plasma Science, Vol. 38, Issue 9
  • DOI: 10.1109/TPS.2010.2049589

A binary collision model for plasma simulation with a particle code
journal, November 1977


Understanding the accuracy of Nanbu’s numerical Coulomb collision operator
journal, July 2009

  • Dimits, Andris M.; Wang, Chiaming; Caflisch, Russel
  • Journal of Computational Physics, Vol. 228, Issue 13
  • DOI: 10.1016/j.jcp.2009.03.041

A high-order finite-volume algorithm for Fokker–Planck collisions in magnetized plasmas
journal, July 2008

  • Xiong, Z.; Cohen, R. H.; Rognlien, T. D.
  • Journal of Computational Physics, Vol. 227, Issue 15
  • DOI: 10.1016/j.jcp.2008.04.004

Linearized model Fokker–Planck collision operators for gyrokinetic simulations. I. Theory
journal, December 2008

  • Abel, I. G.; Barnes, M.; Cowley, S. C.
  • Physics of Plasmas, Vol. 15, Issue 12
  • DOI: 10.1063/1.3046067

Data-parallel algorithms for Monte Carlo simulation of neoclassical transport in magnetically confined plasmas
journal, November 1993


Monte Carlo Transport Simulation Techniques for Stellarator Fusion Experiments
journal, January 1999

  • Dettrick, S. A.; Gardner, H. J.; Painter, S. L.
  • Australian Journal of Physics, Vol. 52, Issue 4
  • DOI: 10.1071/PH98106

Fokker-Planck Equation for an Inverse-Square Force
journal, July 1957

  • Rosenbluth, Marshall N.; MacDonald, William M.; Judd, David L.
  • Physical Review, Vol. 107, Issue 1
  • DOI: 10.1103/PhysRev.107.1

Variable Step Size Control in the Numerical Solution of Stochastic Differential Equations
journal, October 1997


Random Generation of Stochastic Area Integrals
journal, August 1994


Works referencing / citing this record:

A Langevin approach to multi-scale modeling
journal, April 2018


Langevin equation for coulomb collision in non-Maxwellian plasmas
journal, July 2018

  • Oumbarek Espinos, Driss; Zhidkov, Alexei; Kodama, Ryousuke
  • Physics of Plasmas, Vol. 25, Issue 7
  • DOI: 10.1063/1.5025743

Multiscale modelling and splitting approaches for fluids composed of Coulomb-interacting particles
journal, July 2018


A Langevin approach to multi-scale modeling
text, January 2018