Kinetic Simulation of Collisional Magnetized Plasmas with Semi-implicit Time Integration

Ghosh, Debojyoti; Dorf, Mikhail A.; Dorr, Milo R.; Hittinger, Jeffrey A.  F.

doi:10.1007/s10915-018-0726-6

Title: Kinetic Simulation of Collisional Magnetized Plasmas with Semi-implicit Time Integration

Abstract

Plasmas with varying collisionalities occur in many applications, such as tokamak edge regions, where the flows are characterized by significant variations in density and temperature. While a kinetic model is necessary for weakly-collisional high-temperature plasmas, high collisionality in colder regions render the equations numerically stiff due to disparate time scales. In this study, we propose an implicit–explicit algorithm for such cases, where the collisional term is integrated implicitly in time, while the advective term is integrated explicitly in time, thus allowing time step sizes that are comparable to the advective time scales. This partitioning results in a more efficient algorithm than those using explicit time integrators, where the time step sizes are constrained by the stiff collisional time scales. Finally, we implement semi-implicit additive Runge–Kutta methods in COGENT, a high-order finite-volume gyrokinetic code and test the accuracy, convergence, and computational cost of these semi-implicit methods for test cases with highly-collisional plasmas.

Authors:

^[1]; Dorf, Mikhail A. ^[2]; Dorr, Milo R. ^[1]; Hittinger, Jeffrey A. F. ^[1]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Physics Division

Publication Date:: 2018-05-12

Research Org.:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1458696

Report Number(s):: LLNL-JRNL-735522
Journal ID: ISSN 0885-7474; 887762

Grant/Contract Number:: AC52-07NA27344

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Scientific Computing

Additional Journal Information:: Journal Volume: 77; Journal ID: ISSN 0885-7474

Publisher:: Springer

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; IMEX time integration; plasma physics; gyrokinetic simulations; Vlasov–Fokker–Planck equations

Citation Formats


                    Ghosh, Debojyoti, Dorf, Mikhail A., Dorr, Milo R., and Hittinger, Jeffrey A.  F. Kinetic Simulation of Collisional Magnetized Plasmas with Semi-implicit Time Integration.  United States: N. p., 2018. 
Web.  doi:10.1007/s10915-018-0726-6.

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                    Ghosh, Debojyoti, Dorf, Mikhail A., Dorr, Milo R., & Hittinger, Jeffrey A.  F. Kinetic Simulation of Collisional Magnetized Plasmas with Semi-implicit Time Integration.  United States.  https://doi.org/10.1007/s10915-018-0726-6

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                    Ghosh, Debojyoti, Dorf, Mikhail A., Dorr, Milo R., and Hittinger, Jeffrey A.  F. Sat .  
"Kinetic Simulation of Collisional Magnetized Plasmas with Semi-implicit Time Integration".  United States.  https://doi.org/10.1007/s10915-018-0726-6.  https://www.osti.gov/servlets/purl/1458696.

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@article{osti_1458696,

  title        = {Kinetic Simulation of Collisional Magnetized Plasmas with Semi-implicit Time Integration},

  author       = {Ghosh, Debojyoti and Dorf, Mikhail A. and Dorr, Milo R. and Hittinger, Jeffrey A.  F.},

  abstractNote = {Plasmas with varying collisionalities occur in many applications, such as tokamak edge regions, where the flows are characterized by significant variations in density and temperature. While a kinetic model is necessary for weakly-collisional high-temperature plasmas, high collisionality in colder regions render the equations numerically stiff due to disparate time scales. In this study, we propose an implicit–explicit algorithm for such cases, where the collisional term is integrated implicitly in time, while the advective term is integrated explicitly in time, thus allowing time step sizes that are comparable to the advective time scales. This partitioning results in a more efficient algorithm than those using explicit time integrators, where the time step sizes are constrained by the stiff collisional time scales. Finally, we implement semi-implicit additive Runge–Kutta methods in COGENT, a high-order finite-volume gyrokinetic code and test the accuracy, convergence, and computational cost of these semi-implicit methods for test cases with highly-collisional plasmas.},

  doi          = {10.1007/s10915-018-0726-6},

  journal      = {Journal of Scientific Computing},

  number       = ,

  volume       = 77,

  place        = {United States},

  year         = {2018},

  month        = {5}

}

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We propose IMEX HDG-DG schemes for planar and spherical shallow water systems. Of interest is subcritical flow, where the speed of the gravity wave is faster than that of nonlinear advection. In order to simulate these flows efficiently, we split the governing system into a stiff part describing the gravity wave and a non-stiff part associated with nonlinear advection. The former is discretized implicitly with the HDG method while an explicit Runge-Kutta DG discretization is employed for the latter. The proposed IMEX HDG-DG framework: 1) facilitates high-order solutions both in time and space; 2) avoids overly small time-step sizes; 3)more »« less
Cited by 10
https://doi.org/10.1016/j.jcp.2019.109010

Full Text Available
Additive semi-implicit Runge-Kutta methods for computing high-speed nonequilibrium reactive flows

Journal Article Zhong, Xiaolin - Journal of Computational Physics

This paper is concerned with time-stepping numerical methods for computing stiff semi-discrete systems of ordinary differential equations for transient hypersonic flows with thermo-chemical non-equilibrium. The stiffness of the equations is mainly caused by the viscous flux terms across the boundary layers and by the source terms modeling finite-rate thermo-chemical processes. Implicit methods are needed to treat the stiff terms while more efficient explicit methods can still be used for the nonstiff terms in the equations. This paper studies three different semi-implicit Runge-Kutta methods for additively split differential equations in the form of u{prime} = f(u) + g(u), where f ismore »« less
https://doi.org/10.1006/jcph.1996.0193
Evaluation of Implicit‐Explicit Additive Runge‐Kutta Integrators for the HOMME‐NH Dynamical Core

Journal Article Vogl, Christopher J. ; Steyer, Andrew ; Reynolds, Daniel R. ; ... - Journal of Advances in Modeling Earth Systems

Abstract The nonhydrostatic High‐Order Method Modeling Environment (HOMME‐NH) atmospheric dynamical core supports acoustic waves that propagate significantly faster than the advective wind speed, thus greatly limiting the time step size that can be used with standard explicit time integration methods. Resolving acoustic waves is unnecessary for accurate climate and weather prediction. This numerical stiffness is addressed herein by considering implicit‐explicit additive Runge‐Kutta (ARK IMEX) methods that can treat the acoustic waves in a stable manner without requiring implicit treatment of nonstiff modes. Various ARK IMEX methods are evaluated for their efficiency in producing accurate solutions, ability to take large timemore »« less
Cited by 9
https://doi.org/10.1029/2019MS001700

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Title: Kinetic Simulation of Collisional Magnetized Plasmas with Semi-implicit Time Integration

Abstract

Citation Formats

Progress in Kinetic Simulation of Edge Plasmas journal, March 2008

Conservative Numerical Schemes for the Vlasov Equation journal, September 2001

Tempest Simulations of Collisionless Damping of the Geodesic-Acoustic Mode in Edge-Plasma Pedestals journal, May 2008

Detailed comparison of simulated and measured plasma profiles in the scrape-off layer and edge plasma of DIII-D journal, September 2000

Discretization methods for one-dimensional Fokker-Planck operators journal, December 1985

Entropy Stable Numerical Schemes for Two-Fluid Plasma Equations journal, November 2011

An IMEX-Scheme for Pricing Options under Stochastic Volatility Models with Jumps journal, January 2014

Gyrokinetic study of the edge shear layer journal, April 2006

Conservative gyrokinetic Vlasov simulation journal, February 2008

A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations journal, July 2011

Progress with the COGENT Edge Kinetic Code: Collision Operator Options journal, June 2012

Relaxation of a system of charged particles journal, July 1985

TEMPEST simulations of the plasma transport in a single-null tokamak geometry journal, May 2010

A drift-kinetic Semi-Lagrangian 4D code for ion turbulence simulation journal, September 2006

Fast implicit schemes for the Fokker–Planck–Landau equation journal, May 2004

A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth–Fokker–Planck equation journal, September 2015

Nonlinear gyrokinetic equations for turbulence in core transport barriers journal, December 1996

Additive Runge–Kutta schemes for convection–diffusion–reaction equations journal, January 2003

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

Edge gyrokinetic theory and continuum simulations journal, July 2007

NITSOL: A Newton Iterative Solver for Nonlinear Systems journal, January 1998

Implicit–explicit time integration of a high-order particle-in-cell method with hyperbolic divergence cleaning journal, October 2009

High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids journal, May 2015

Fast Algorithms for Numerical, Conservative, and Entropy Approximations of the Fokker–Planck–Landau Equation journal, May 1997

A forward semi-Lagrangian method for the numerical solution of the Vlasov equation journal, October 2009

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

New velocity-space discretization for continuum kinetic calculations and Fokker–Planck collisions journal, June 2013

An implicit Vlasov–Fokker–Planck code to model non-local electron transport in 2-D with magnetic fields journal, February 2004

Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation journal, November 2005

Kinetic simulation of a plasma collision experiment journal, August 1993

Iterative Methods for Sparse Linear Systems book, January 2003

GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems journal, July 1986

Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov–Poisson system journal, September 2011

Implicit–explicit Runge–Kutta methods applied to atmospheric chemistry-transport modelling journal, September 2000

An energy- and charge-conserving, nonlinearly implicit, electromagnetic 1D-3V Vlasov–Darwin particle-in-cell algorithm journal, October 2014

Particle Simulation of the Neoclassical Plasmas journal, November 2001

Conservative finite-difference schemes for the Fokker-Planck equation not violating the law of an increasing entropy journal, March 1987

Simulation of neoclassical transport with the continuum gyrokinetic code COGENT journal, January 2013

Fully Implicit Kinetic Solution of Collisional Plasmas journal, September 1997

Spontaneous rotation sources in a quiescent tokamak edge plasma journal, June 2008

Full linearized Fokker–Planck collisions in neoclassical transport simulations journal, December 2011

Progress with the COGENT Edge Kinetic Code: Implementing the Fokker-Planck Collision Operator: Progress with the COGENT Edge Kinetic Code: Implementing the Fokker-Planck Collision Operator journal, June 2014

Fast elliptic solvers in cylindrical coordinates and the Coulomb collision operator journal, September 2011

Kinetic simulation of a collisional shock wave in a plasma journal, October 1991

Conservative and Entropy Decaying Numerical Scheme for the Isotropic Fokker–Planck–Landau Equation journal, September 1998

Semi-Implicit Formulations of the Navier–Stokes Equations: Application to Nonhydrostatic Atmospheric Modeling journal, January 2010

An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes journal, August 2016

An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm journal, August 2011

Kinetic simulations of fuel ion transport in ICF target implosions journal, November 2003

FPPAC: A two-dimensional multispecies nonlinear Fokker-Planck package journal, September 1981

A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov–Darwin particle-in-cell algorithm journal, December 2015

The solution of poisson's equation for isolated source distributions journal, October 1977

hypre: A Library of High Performance Preconditioners book, April 2002

Implicit and Conservative Difference Scheme for the Fokker-Planck Equation journal, June 1994

Gyrokinetic Simulation of Particle and Heat Transport in the Presence of Wide Orbits and Strong Profile Variations in the Edge Plasma journal, September 2006

Gyrokinetic approach in particle simulation journal, January 1983

A hybrid method of semi-Lagrangian and additive semi-implicit Runge–Kutta schemes for gyrokinetic Vlasov simulations journal, September 2012

Implicit–Explicit Multistep Methods for Fast-Wave–Slow-Wave Problems journal, April 2012

Conservative global gyrokinetic toroidal full-f five-dimensional Vlasov simulation journal, September 2008

An Implicit Energy-Conservative 2D Fokker–Planck Algorithm journal, January 2000

Implicit-Explicit Formulations of a Three-Dimensional Nonhydrostatic Unified Model of the Atmosphere (NUMA) journal, January 2013

Two-Dimensional Nonlocal Electron Transport in Laser-Produced Plasmas journal, November 1988

Vlasov simulations of electron-ion collision effects on damping of electron plasma waves journal, March 2016

A second order self-consistent IMEX method for radiation hydrodynamics journal, November 2010

Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations journal, November 1997

A review of Vlasov–Fokker–Planck numerical modeling of inertial confinement fusion plasma journal, February 2012

Implicit Schemes for the Fokker--Planck--Landau Equation journal, January 2005

Rapid self-magnetization of laser speckles in plasmas by nonlinear anisotropic instability journal, March 2009

Semi-implicit Time Integration of Atmospheric Flows with Characteristic-Based Flux Partitioning journal, January 2016

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

Multipole expansions for the Fokker-Planck-Landau operator journal, February 1998

A Numerical Method for the Accurate Solution of the Fokker–Planck–Landau Equation in the Nonhomogeneous Case journal, June 2002

High-order, finite-volume methods in mapped coordinates journal, April 2011

A conservative high order semi-Lagrangian WENO method for the Vlasov equation journal, February 2010

A practical difference scheme for Fokker-Planck equations journal, August 1970

An Implicit Energy-Conservative 2D Fokker–Planck Algorithm journal, January 2000

FPPAC: A two-dimensional multispecies nonlinear fokker-planck package journal, January 1984

Progress in Kinetic Simulation of Edge Plasmas
journal, March 2008

Conservative Numerical Schemes for the Vlasov Equation
journal, September 2001

Tempest Simulations of Collisionless Damping of the Geodesic-Acoustic Mode in Edge-Plasma Pedestals
journal, May 2008

Detailed comparison of simulated and measured plasma profiles in the scrape-off layer and edge plasma of DIII-D
journal, September 2000

Discretization methods for one-dimensional Fokker-Planck operators
journal, December 1985

Entropy Stable Numerical Schemes for Two-Fluid Plasma Equations
journal, November 2011

An IMEX-Scheme for Pricing Options under Stochastic Volatility Models with Jumps
journal, January 2014

Gyrokinetic study of the edge shear layer
journal, April 2006

Conservative gyrokinetic Vlasov simulation
journal, February 2008

A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations
journal, July 2011

Progress with the COGENT Edge Kinetic Code: Collision Operator Options
journal, June 2012

Relaxation of a system of charged particles
journal, July 1985

TEMPEST simulations of the plasma transport in a single-null tokamak geometry
journal, May 2010

A drift-kinetic Semi-Lagrangian 4D code for ion turbulence simulation
journal, September 2006

Fast implicit schemes for the Fokker–Planck–Landau equation
journal, May 2004

A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth–Fokker–Planck equation
journal, September 2015

Nonlinear gyrokinetic equations for turbulence in core transport barriers
journal, December 1996

Additive Runge–Kutta schemes for convection–diffusion–reaction equations
journal, January 2003

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

Edge gyrokinetic theory and continuum simulations
journal, July 2007

NITSOL: A Newton Iterative Solver for Nonlinear Systems
journal, January 1998

Implicit–explicit time integration of a high-order particle-in-cell method with hyperbolic divergence cleaning
journal, October 2009

High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids
journal, May 2015

Fast Algorithms for Numerical, Conservative, and Entropy Approximations of the Fokker–Planck–Landau Equation
journal, May 1997

A forward semi-Lagrangian method for the numerical solution of the Vlasov equation
journal, October 2009

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

New velocity-space discretization for continuum kinetic calculations and Fokker–Planck collisions
journal, June 2013

An implicit Vlasov–Fokker–Planck code to model non-local electron transport in 2-D with magnetic fields
journal, February 2004

Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation
journal, November 2005

Kinetic simulation of a plasma collision experiment
journal, August 1993

Iterative Methods for Sparse Linear Systems
book, January 2003

GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
journal, July 1986

Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov–Poisson system
journal, September 2011

Implicit–explicit Runge–Kutta methods applied to atmospheric chemistry-transport modelling
journal, September 2000

An energy- and charge-conserving, nonlinearly implicit, electromagnetic 1D-3V Vlasov–Darwin particle-in-cell algorithm
journal, October 2014

Particle Simulation of the Neoclassical Plasmas
journal, November 2001

Conservative finite-difference schemes for the Fokker-Planck equation not violating the law of an increasing entropy
journal, March 1987

Simulation of neoclassical transport with the continuum gyrokinetic code COGENT
journal, January 2013

Fully Implicit Kinetic Solution of Collisional Plasmas
journal, September 1997

Spontaneous rotation sources in a quiescent tokamak edge plasma
journal, June 2008

Full linearized Fokker–Planck collisions in neoclassical transport simulations
journal, December 2011

Progress with the COGENT Edge Kinetic Code: Implementing the Fokker-Planck Collision Operator: Progress with the COGENT Edge Kinetic Code: Implementing the Fokker-Planck Collision Operator
journal, June 2014

Fast elliptic solvers in cylindrical coordinates and the Coulomb collision operator
journal, September 2011

Kinetic simulation of a collisional shock wave in a plasma
journal, October 1991

Conservative and Entropy Decaying Numerical Scheme for the Isotropic Fokker–Planck–Landau Equation
journal, September 1998

Semi-Implicit Formulations of the Navier–Stokes Equations: Application to Nonhydrostatic Atmospheric Modeling
journal, January 2010

An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes
journal, August 2016

An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm
journal, August 2011

Kinetic simulations of fuel ion transport in ICF target implosions
journal, November 2003

FPPAC: A two-dimensional multispecies nonlinear Fokker-Planck package
journal, September 1981

A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov–Darwin particle-in-cell algorithm
journal, December 2015

The solution of poisson's equation for isolated source distributions
journal, October 1977

hypre: A Library of High Performance Preconditioners
book, April 2002

Implicit and Conservative Difference Scheme for the Fokker-Planck Equation
journal, June 1994

Gyrokinetic Simulation of Particle and Heat Transport in the Presence of Wide Orbits and Strong Profile Variations in the Edge Plasma
journal, September 2006

Gyrokinetic approach in particle simulation
journal, January 1983

A hybrid method of semi-Lagrangian and additive semi-implicit Runge–Kutta schemes for gyrokinetic Vlasov simulations
journal, September 2012

Implicit–Explicit Multistep Methods for Fast-Wave–Slow-Wave Problems
journal, April 2012

Conservative global gyrokinetic toroidal full-f five-dimensional Vlasov simulation
journal, September 2008

An Implicit Energy-Conservative 2D Fokker–Planck Algorithm
journal, January 2000

Implicit-Explicit Formulations of a Three-Dimensional Nonhydrostatic Unified Model of the Atmosphere (NUMA)
journal, January 2013

Two-Dimensional Nonlocal Electron Transport in Laser-Produced Plasmas
journal, November 1988

Vlasov simulations of electron-ion collision effects on damping of electron plasma waves
journal, March 2016

A second order self-consistent IMEX method for radiation hydrodynamics
journal, November 2010

Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
journal, November 1997

A review of Vlasov–Fokker–Planck numerical modeling of inertial confinement fusion plasma
journal, February 2012

Implicit Schemes for the Fokker--Planck--Landau Equation
journal, January 2005

Rapid self-magnetization of laser speckles in plasmas by nonlinear anisotropic instability
journal, March 2009

Semi-implicit Time Integration of Atmospheric Flows with Characteristic-Based Flux Partitioning
journal, January 2016

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

Multipole expansions for the Fokker-Planck-Landau operator
journal, February 1998

A Numerical Method for the Accurate Solution of the Fokker–Planck–Landau Equation in the Nonhomogeneous Case
journal, June 2002

High-order, finite-volume methods in mapped coordinates
journal, April 2011

A conservative high order semi-Lagrangian WENO method for the Vlasov equation
journal, February 2010

A practical difference scheme for Fokker-Planck equations
journal, August 1970

An Implicit Energy-Conservative 2D Fokker–Planck Algorithm
journal, January 2000

FPPAC: A two-dimensional multispecies nonlinear fokker-planck package
journal, January 1984

A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations
text, January 2010

An Energy- and Charge-conserving, Implicit, Electrostatic Particle-in-Cell Algorithm
text, January 2011

New velocity-space discretization for continuum kinetic calculations and Fokker-Planck collisions
text, January 2012