An equilibrium-preserving discretization for the nonlinear Rosenbluth–Fokker–Planck operator in arbitrary multi-dimensional geometry

Taitano, William T.; Chacon, Luis; Simakov, Andrei Nikolaevich

doi:10.1016/j.jcp.2017.03.032

An equilibrium-preserving discretization for the nonlinear Rosenbluth–Fokker–Planck operator in arbitrary multi-dimensional geometry

Journal Article · Thu Apr 06 00:00:00 EDT 2017 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2017.03.032· OSTI ID:1460628

^[1]; ^[1]; ^[1]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Here, the Fokker–Planck collision operator is an advection-diffusion operator which describe dynamical systems such as weakly coupled plasmas, photonics in high temperature environment biological, and even social systems. For plasmas in the continuum, the Fokker–Planck collision operator supports such important physical properties as conservation of number, momentum, and energy, as well as positivity. It also obeys the Boltzmann's H-theorem, i.e., the operator increases the system entropy while simultaneously driving the distribution function towards a Maxwellian. In the discrete, when these properties are not ensured, numerical simulations can either fail catastrophically or suffer from significant numerical pollution. There is strong emphasis in the literature on developing numerical techniques to solve the Fokker–Planck equation while preserving these properties. In this short note, we focus on the analytical equilibrium preserving property, meaning that the Fokker–Planck collision operator vanishes when acting on an analytical Maxwellian distribution function. The equilibrium preservation property is especially important, for example, when one is attempting to capture subtle transport physics. Since transport arises from small Ο(ϵ) corrections to the equilibrium (where ϵ is a small expansion parameter), numerical truncation error present in the equilibrium solution may dominate, overwhelming transport dynamics.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC52-06NA25396

OSTI ID:: 1460628

Alternate ID(s):: OSTI ID: 1396735

Report Number(s):: LA-UR--16-28802

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 339; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (17)

Curvature-compensated convective transport: SMART, A new boundedness- preserving transport algorithm Gaskell, P. H.; Lau, A. K. C. International Journal for Numerical Methods in Fluids, Vol. 8, Issue 6 https://doi.org/10.1002/fld.1650080602	journal	June 1988
Implicit and Conservative Difference Scheme for the Fokker-Planck Equation Epperlein, E. M. Journal of Computational Physics, Vol. 112, Issue 2 https://doi.org/10.1006/jcph.1994.1101	journal	June 1994
An Implicit Energy-Conservative 2D Fokker–Planck Algorithm Chacón, L.; Barnes, D. C.; Knoll, D. A. Journal of Computational Physics, Vol. 157, Issue 2 https://doi.org/10.1006/jcph.1999.6394	journal	January 2000
A Numerical Method for the Accurate Solution of the Fokker–Planck–Landau Equation in the Nonhomogeneous Case Filbet, Francis; Pareschi, Lorenzo Journal of Computational Physics, Vol. 179, Issue 1 https://doi.org/10.1006/jcph.2002.7010	journal	June 2002
On boltzmann equations and fokker—planck asymptotics: Influence of grazing collisions Goudon, T. Journal of Statistical Physics, Vol. 89, Issue 3-4 https://doi.org/10.1007/BF02765543	journal	November 1997
An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory Degond, Pierre; Lucquin-Desreux, Brigitte Numerische Mathematik, Vol. 68, Issue 2 https://doi.org/10.1007/s002110050059	journal	July 1994
A practical difference scheme for Fokker-Planck equations Chang, J. S.; Cooper, G. Journal of Computational Physics, Vol. 6, Issue 1 https://doi.org/10.1016/0021-9991(70)90001-X	journal	August 1970
Discretization methods for one-dimensional Fokker-Planck operators Larsen, E. W.; Levermore, C. D.; Pomraning, G. C. Journal of Computational Physics, Vol. 61, Issue 3 https://doi.org/10.1016/0021-9991(85)90070-1	journal	December 1985
On the existence of a generalized solution of Landau's equation Arsen'ev, A. A.; Peskov, N. V. USSR Computational Mathematics and Mathematical Physics, Vol. 17, Issue 4 https://doi.org/10.1016/0041-5553(77)90125-2	journal	January 1977
Conservative difference schemes for the Fokker-Planck equation Pekker, M. S.; Khudik, V. N. USSR Computational Mathematics and Mathematical Physics, Vol. 24, Issue 3 https://doi.org/10.1016/0041-5553(84)90075-2	journal	January 1984
A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources Filbet, Francis; Jin, Shi Journal of Computational Physics, Vol. 229, Issue 20 https://doi.org/10.1016/j.jcp.2010.06.017	journal	October 2010
A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth–Fokker–Planck equation Taitano, W. T.; Chacón, L.; Simakov, A. N. Journal of Computational Physics, Vol. 297 https://doi.org/10.1016/j.jcp.2015.05.025	journal	September 2015
An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes Taitano, W. T.; Chacón, L.; Simakov, A. N. Journal of Computational Physics, Vol. 318 https://doi.org/10.1016/j.jcp.2016.03.071	journal	August 2016
On asymptotics of the Boltzmann equation when the collisions become grazing Desvillettes, L. Transport Theory and Statistical Physics, Vol. 21, Issue 3 https://doi.org/10.1080/00411459208203923	journal	June 1992
Fokker-Planck Equation for an Inverse-Square Force Rosenbluth, Marshall N.; MacDonald, William M.; Judd, David L. Physical Review, Vol. 107, Issue 1 https://doi.org/10.1103/PhysRev.107.1	journal	July 1957
The Fokker-Planck Asymptotics of the Boltzmann Collision Operator in the Coulomb case Degond, P.; Lucquin-Desreux, B. Mathematical Models and Methods in Applied Sciences, Vol. 02, Issue 02 https://doi.org/10.1142/S0218202592000119	journal	June 1992
Minimal stencil finite volume scheme with the discrete maximum principle Lipnikov, K.; Svyatskiy, D.; Vassilevski, Yu. Russian Journal of Numerical Analysis and Mathematical Modelling, Vol. 27, Issue 4 https://doi.org/10.1515/rnam-2012-0020	journal	January 2012

Cited By (9)

Plasma ion stratification by weak planar shocks Simakov, Andrei N.; Keenan, Brett D.; Taitano, William T. Physics of Plasmas, Vol. 24, Issue 9 https://doi.org/10.1063/1.4995427	journal	September 2017
Ion species stratification within strong shocks in two-ion plasmas Keenan, Brett D.; Simakov, Andrei N.; Taitano, William T. Physics of Plasmas, Vol. 25, Issue 3 https://doi.org/10.1063/1.5020156	journal	March 2018
Yield degradation in inertial-confinement-fusion implosions due to shock-driven kinetic fuel-species stratification and viscous heating Taitano, W. T.; Simakov, A. N.; Chacón, L. Physics of Plasmas, Vol. 25, Issue 5 https://doi.org/10.1063/1.5024402	journal	May 2018
Diffusion-driven fluid dynamics in ideal gases and plasmas Vold, E. L.; Yin, L.; Taitano, W. Physics of Plasmas, Vol. 25, Issue 6 https://doi.org/10.1063/1.5029932	journal	June 2018
Kinetic physics in ICF: present understanding and future directions Rinderknecht, Hans G.; Amendt, P. A.; Wilks, S. C. Plasma Physics and Controlled Fusion, Vol. 60, Issue 6 https://doi.org/10.1088/1361-6587/aab79f	journal	April 2018
Conservative discrete-velocity method for the ellipsoidal Fokker-Planck equation in gas-kinetic theory Liu, Sha; Yuan, Ruifeng; Javid, Usman Physical Review E, Vol. 100, Issue 3 https://doi.org/10.1103/physreve.100.033310	journal	September 2019
Deciphering the kinetic structure of multi-ion plasma shocks Keenan, Brett D.; Simakov, Andrei N.; Chacón, Luis Physical Review E, Vol. 96, Issue 5 https://doi.org/10.1103/physreve.96.053203	journal	November 2017
Deciphering the Kinetic Structure of Multi-Ion Plasma Shocks Keenan, Brett D.; Simakov, Andrei N.; Chacon, Luis arXiv https://doi.org/10.48550/arxiv.1707.02927	text	January 2017
Ion Species Stratification Within Strong Shocks in Two-Ion Plasmas Keenan, Brett D.; Simakov, Andrei N.; Taitano, William T. arXiv https://doi.org/10.48550/arxiv.1712.08663	text	January 2017

Similar Records

Moment-Preserving SN Discretizations for the One-Dimensional Fokker-Planck Equation

Conference · Thu Jun 14 00:00:00 EDT 2012 · OSTI ID:1044079

Fast algorithms for numerical, conservative, and entropy approximations of the Fokker-Planck-Landau equation

Journal Article · Thu May 15 00:00:00 EDT 1997 · Journal of Computational Physics · OSTI ID:535388

Radial transport/Fokker-Planck model for a tokamak plasma

Journal Article · Fri Dec 31 23:00:00 EST 1976 · J. Comput. Phys.; (United States) · OSTI ID:7212544

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
Conservative discretization
Exact analytical equilibrium preserving
Fokker–Planck
Mathematics
Multiple dimensions
Rosenbluth potentials

An equilibrium-preserving discretization for the nonlinear Rosenbluth–Fokker–Planck operator in arbitrary multi-dimensional geometry

Citation Formats

References (17)

Cited By (9)

Similar Records

Related Subjects