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

Title: Simulating pitch angle scattering using an explicitly solvable energy-conserving algorithm

Abstract

Particle distribution functions evolving under the Lorentz operator can be simulated with the Langevin equation for pitch angle scattering. This approach is frequently used in particle based Monte-Carlo simulations of plasma collisions, among others. However, most numerical treatments do not guarantee energy conservation, which may lead to unphysical artifacts such as numerical heating and spectra distortions. In this paper, we present a novel structure-preserving numerical algorithm for the Langevin equation for pitch angle scattering. Similar to the well-known Boris algorithm, the proposed numerical scheme takes advantage of the structure-preserving properties of the Cayley transform when calculating the velocity-space rotations. The resulting algorithm is explicitly solvable, while preserving the norm of velocities down to machine precision. We demonstrate that the method has the same order of numerical convergence as the traditional stochastic Euler-Maruyama method. The numerical scheme is benchmarked by simulating the pitch angle scattering of a particle beam, and comparing with the analytical solution. Benchmark results show excellent agreement with theoretical predictions, showcasing the remarkable long time accuracy of the proposed algorithm.

Authors:
ORCiD logo [1]; ORCiD logo [1]; ORCiD logo [1]
  1. Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States); Princeton Univ., NJ (United States)
Publication Date:
Research Org.:
Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1660496
Grant/Contract Number:  
AC02-09CH11466
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Physical Review E
Additional Journal Information:
Journal Volume: 102; Journal Issue: 3; Journal ID: ISSN 2470-0045
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English

Citation Formats

Zhang, Xin, Fu, Yichen, and Qin, Hong. Simulating pitch angle scattering using an explicitly solvable energy-conserving algorithm. United States: N. p., 2020. Web. doi:10.1103/physreve.102.033302.
Zhang, Xin, Fu, Yichen, & Qin, Hong. Simulating pitch angle scattering using an explicitly solvable energy-conserving algorithm. United States. doi:10.1103/physreve.102.033302.
Zhang, Xin, Fu, Yichen, and Qin, Hong. Tue . "Simulating pitch angle scattering using an explicitly solvable energy-conserving algorithm". United States. doi:10.1103/physreve.102.033302.
@article{osti_1660496,
title = {Simulating pitch angle scattering using an explicitly solvable energy-conserving algorithm},
author = {Zhang, Xin and Fu, Yichen and Qin, Hong},
abstractNote = {Particle distribution functions evolving under the Lorentz operator can be simulated with the Langevin equation for pitch angle scattering. This approach is frequently used in particle based Monte-Carlo simulations of plasma collisions, among others. However, most numerical treatments do not guarantee energy conservation, which may lead to unphysical artifacts such as numerical heating and spectra distortions. In this paper, we present a novel structure-preserving numerical algorithm for the Langevin equation for pitch angle scattering. Similar to the well-known Boris algorithm, the proposed numerical scheme takes advantage of the structure-preserving properties of the Cayley transform when calculating the velocity-space rotations. The resulting algorithm is explicitly solvable, while preserving the norm of velocities down to machine precision. We demonstrate that the method has the same order of numerical convergence as the traditional stochastic Euler-Maruyama method. The numerical scheme is benchmarked by simulating the pitch angle scattering of a particle beam, and comparing with the analytical solution. Benchmark results show excellent agreement with theoretical predictions, showcasing the remarkable long time accuracy of the proposed algorithm.},
doi = {10.1103/physreve.102.033302},
journal = {Physical Review E},
issn = {2470-0045},
number = 3,
volume = 102,
place = {United States},
year = {2020},
month = {9}
}

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on September 1, 2021
Publisher's Version of Record

Save / Share:

Works referenced in this record:

Simulation of runaway electrons during tokamak disruptions
journal, August 2003


Metriplectic integrators for the Landau collision operator
journal, October 2017

  • Kraus, Michael; Hirvijoki, Eero
  • Physics of Plasmas, Vol. 24, Issue 10
  • DOI: 10.1063/1.4998610

High Order Conformal Symplectic and Ergodic Schemes for the Stochastic Langevin Equation via Generating Functions
journal, January 2017

  • Hong, Jialin; Sun, Liying; Wang, Xu
  • SIAM Journal on Numerical Analysis, Vol. 55, Issue 6
  • DOI: 10.1137/17M111691X

Langevin approach to plasma kinetics with Coulomb collisions
journal, January 1999


Collisionless pitch-angle scattering of runaway electrons
journal, May 2016


Numerical Methods for Stochastic Systems Preserving Symplectic Structure
journal, January 2002

  • Milstein, G. N.; Repin, Yu. M.; Tretyakov, M. V.
  • SIAM Journal on Numerical Analysis, Vol. 40, Issue 4
  • DOI: 10.1137/S0036142901395588

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


A Can0nical Integrati0n Technique
journal, August 1983


Explicit high-order gauge-independent symplectic algorithms for relativistic charged particle dynamics
journal, August 2019


Monte Carlo simulation of runaway electrons in a toroidal geometry
journal, July 1993

  • Heikkinen, J. A.; Sipilä, S. K.; Pättikangas, T. J. H.
  • Computer Physics Communications, Vol. 76, Issue 2
  • DOI: 10.1016/0010-4655(93)90133-W

From nonlinear Fokker–Planck equations to solutions of distribution dependent SDE
journal, July 2020

  • Barbu, Viorel; Röckner, Michael
  • Annals of Probability, Vol. 48, Issue 4
  • DOI: 10.1214/19-AOP1410

High order volume-preserving algorithms for relativistic charged particles in general electromagnetic fields
journal, September 2016

  • He, Yang; Sun, Yajuan; Zhang, Ruili
  • Physics of Plasmas, Vol. 23, Issue 9
  • DOI: 10.1063/1.4962677

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

Why is Boris algorithm so good?
journal, August 2013

  • Qin, Hong; Zhang, Shuangxi; Xiao, Jianyuan
  • Physics of Plasmas, Vol. 20, Issue 8
  • DOI: 10.1063/1.4818428

Runge-kutta schemes for Hamiltonian systems
journal, December 1988


Symplectic Integration of Hamiltonian Systems with Additive Noise
journal, January 2002

  • Milstein, G. N.; Repin, Yu. M.; Tretyakov, M. V.
  • SIAM Journal on Numerical Analysis, Vol. 39, Issue 6
  • DOI: 10.1137/S0036142901387440

A computational investigation of the random particle method for numerical solution of the kinetic Vlasov-Poisson-Fokker-Planck equations
journal, September 1994


Stochastic symplectic Runge–Kutta methods for the strong approximation of Hamiltonian systems with additive noise
journal, December 2017

  • Zhou, Weien; Zhang, Jingjing; Hong, Jialin
  • Journal of Computational and Applied Mathematics, Vol. 325
  • DOI: 10.1016/j.cam.2017.04.050

On the Theory of the Brownian Motion
journal, September 1930


Multilevel Monte Carlo simulation of Coulomb collisions
journal, October 2014


Distribution dependent SDEs for Landau type equations
journal, February 2018


Geometric Euler--Maruyama Schemes for Stochastic Differential Equations in SO(n) and SE(n)
journal, January 2016

  • Piggott, M. J.; Solo, V.
  • SIAM Journal on Numerical Analysis, Vol. 54, Issue 4
  • DOI: 10.1137/15M1019726

Stochastic discrete Hamiltonian variational integrators
journal, August 2018


Midpoint numerical technique for stochastic Landau-Lifshitz-Gilbert dynamics
journal, April 2006

  • d’Aquino, M.; Serpico, C.; Coppola, G.
  • Journal of Applied Physics, Vol. 99, Issue 8
  • DOI: 10.1063/1.2169472

Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems
journal, December 2016


Generating functions for stochastic symplectic methods
journal, January 2014


Explicit K -symplectic algorithms for charged particle dynamics
journal, February 2017


Volume-preserving algorithms for charged particle dynamics
journal, January 2015


Higher order volume-preserving schemes for charged particle dynamics
journal, January 2016


Higher-order time integration of Coulomb collisions in a plasma using Langevin equations
journal, June 2013


Geometrical integration of Landau–Lifshitz–Gilbert equation based on the mid-point rule
journal, November 2005

  • d’Aquino, Massimiliano; Serpico, Claudio; Miano, Giovanni
  • Journal of Computational Physics, Vol. 209, Issue 2
  • DOI: 10.1016/j.jcp.2005.04.001