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A Unified Gas Kinetic Scheme for Multi-scale and Multi-component Plasma Transport

Chang Liu, Kun Xu

arXiv:1609.05291v2physics.comp-phmath.NAphysics.plasm-ph

TL;DR

The paper addresses the difficulty of simulating multi-species plasma across kinetic, transitional, and hydrodynamic regimes with existing specialized methods. It develops a UGKS from coupled Vlasov-BGK and Maxwell equations, recovering these regimes through scale and parameter variation. Tests report smooth transitions from neutral-fluid to MHD and from kinetic Vlasov to hydrodynamic two-fluid results.

  • Problem

    Existing methods have limitations across plasma regimes, including transitional-regime reliability and schemes unable to resolve two-species effects or the MHD limit.

  • Method

    UGKS couples Vlasov-BGK equations for electrons and ions with the full Maxwell equations, incorporating collisions, transport, acceleration, and implicit source treatment.

  • Results

    UGKS produces smooth transitions from neutral-fluid results to MHD solutions and from the kinetic Vlasov equation to the hydrodynamic two-fluid system.

  • Takeaways & Limitations

    The resulting multi-scale method covers plasma flow from the collisionless Vlasov regime to hydrodynamic two-fluid equations.

  • Takeaways & Limitations

    The cited introduction states that hybrid methods may be physically unreliable in transitional regimes and that some existing schemes cannot resolve two-species effects or the MHD limit.

Abstract

from arXiv · show

A unified gas kinetic scheme (UGKS) for multi-scale and multi-component plasma transport is constructed. The current scheme is a direct modeling method, where the time evolution solutions from the Vlasov-BGK equations of electron and ion and the Maxwell equations are used to construct a scale-dependent plasma simulation model. As a result, with the changing of modeling scales of mesh size and time step and with a variation of Knudsen number and Larmor radius, the discretized governing equations for a wide range of plasma evolution regimes can be obtained. The physics recovered in UGKS ranges from the Vlasov equation in the kinetic scale to different-type magneto-hydrodynamic (MHD) equations in the hydrodynamic scale. The key dynamics in UGKS is the un-splitting treatment of particle collision, acceleration, and transport in the construction of numerical flux across a cell interface. At the same time, the plasma evolution is coupled with the Maxwell equations in an implicit way, which automatically provides a smooth transition between the Ampere's law and the Ohm's law for the calculation of electric field. The time step of UGKS is not limited by the relaxation time, the cyclotron period, and the speed of light in the MHD regime. The UGKS is validated by numerical test cases, such as the Landau damping and two stream instability in the kinetic regime, and the Brio-Wu shock tube problem and the Orszag- Tang MHD turbulence problem in the hydrodynamic regime. The scheme is also used to study the geospace environment modeling (GEM), such as the challenging magnetic reconnection problem in the transition regime. Overall, the UGKS is a physically reliable multi-scale plasma simulation method. It provides a powerful and unified approach for the study of plasma physics.

1. Introduction

Plasma flow spans kinetic, transitional, and hydrodynamic regimes governed by collision intensity and magnetization. Existing methods are specialized or limited across regimes, motivating a unified gas kinetic scheme based on coupled kinetic and Maxwell equations.

  • Proposed approach: The paper proposes a UGKS for fully ionized weakly coupled plasma using Vlasov-BGK evolution for electrons and ions together with Maxwell equations.The scheme constructs numerical fluxes from coupled transport, collision, and acceleration, with implicit source-term treatment.
  • Flow regimes: The Knudsen number measures collision intensity, while the normalized Larmor radius characterizes plasma magnetization.These parameters organize the relevant plasma flow regimes.
  • Flow regimes: Plasma behavior ranges from collisionless Vlasov dynamics at large Knudsen number to hydrodynamic equations in highly collisional regimes.Strong magnetization can yield single-fluid MHD, while two-fluid or Hall effects become important in other regimes.
  • Methodological gap: Existing plasma solvers are specialized: PIC methods suffer statistical noise and restricted time steps, while Vlasov solvers primarily target collisionless dynamics.Hybrid methods also face difficulty selecting coupling criteria and may be physically unreliable in transitional regimes.
  • Methodological gap: Single-fluid hydrodynamic solvers are limited to scales much larger than the Larmor radius and regimes close to equilibrium.Some asymptotic-preserving schemes are built on one species and cannot resolve two-species effects or the MHD limit.
  • Proposed approach: By varying mesh and time scales together with Knudsen number, inter-species collision frequency, and normalized Larmor radius, UGKS connects Vlasov, two-fluid, and MHD regimes smoothly.The stated goal is physically accurate plasma simulation across large parameter variations.

2. A review of kinetic and hydrodynamic model equations of plasma

The BGK-Maxwell formulation couples electron and ion kinetic equations with Maxwell equations, covering plasma behavior from collisionless Vlasov dynamics to continuum MHD regimes. Its asymptotic limits recover two-fluid, Euler, resistive-MHD, Hall-MHD, and ideal-MHD descriptions under specified parameter limits.

  • Kinetic and Maxwell equations: The model uses kinetic equations for electrons and ions with averaged electromagnetic forces, while electric and magnetic fields follow Maxwell equations.The formulation includes divergence constraints and perfectly hyperbolic Maxwell equations for electromagnetic-field evolution.
  • Collision modeling: A single AAP BGK collision operator represents both self-collision and cross-collision for each plasma component.The scheme can also incorporate other collision operators, including the Landau operator, using the proposed Boltzmann-collision treatment.
  • Similarity parameters: The BGK-Maxwell system spans flow regimes from collisionless Vlasov dynamics to continuum MHD behavior.Its governing equations are parameterized by normalized relaxation time, Debye length, Larmor radius, speed of light, and electron mass.
  • Hydrodynamic limits: When the relaxation time is small, the zeroth-order system recovers hydrodynamic two-fluid equations with electron-ion momentum and energy exchange.The relaxation time is proportional to the Knudsen number, linking this limit to collisional plasma behavior.
  • Hydrodynamic limits: Intensive interspecies interactions produce Euler equations for total density, common velocity, and common temperature.Further limits in Larmor radius, mass ratio, and scaled Debye length yield MHD equations with Hall effect and magnetic diffusion.
  • MHD limits: The continuum hierarchy includes resistive-MHD, Hall-MHD, and ideal-MHD equations, with the ideal-MHD system obtained as a zeroth-order Larmor-radius approximation.The formulation also applies in transition regimes with moderate Knudsen number, Debye length, and Larmor radius.

3. Unified gas kinetic scheme

UGKS evolves phase-space distribution functions, conservative variables, and electromagnetic fields in a finite-volume framework. Its interface flux is constructed from time-dependent kinetic solutions that include transport, Lorentz acceleration, collision, and reconstructed local data.

  • General framework: UGKS is a finite-volume scheme defined on phase space and couples distribution-function, conservative-variable, and electromagnetic-field evolution.Numerical fluxes for distribution functions and conservative variables are obtained from the time-dependent interface distribution.
  • Numerical flux: The kinetic equation is solved through an integral solution that transports particles and accounts for acceleration due to the Lorentz force.The interface is represented in a local basis, with reconstructed initial distributions around the cell interface and velocity-cell center.
  • Numerical flux: Spatial and velocity slopes are reconstructed in phase-space cells, using a limiter such as van Leer for the initial distribution.The reconstructed distribution supplies interface macroscopic variables and their derivatives.
  • Numerical flux: The post-collision distribution is expanded at the interface with left-right spatial slopes, velocity slopes, and a time-dependent coefficient.These coefficients are used to construct the time-dependent interface distribution and resulting UGKS flux.
  • Field evolution: The electromagnetic field update uses a perfectly hyperbolic Maxwell formulation and wave-propagation fluxes for the field variables.The flowchart summarizes the UGKS algorithm, including numerical flux and field-update stages.
  • Numerical update: A two-step update shifts the velocity distribution between cell centers and then advances it to the next time step.The velocity shift is based on electromagnetic acceleration, with a substep selected from acceleration-induced velocity-cell crossing constraints.

4. Limiting solutions of UGKS

UGKS recovers kinetic and hydrodynamic flux limits according to the ratio of time step to relaxation time. Its implicit source treatment preserves plasma asymptotics while relaxing cyclotron-period and light-speed time-step constraints in continuum regimes.

  • Flux limits: When ∆t ≪τ, the UGKS flux follows particle transport and acceleration, consistent with the collisionless Vlasov equation.When ∆t ≫τ, the flux converges to hydrodynamic flux instead.
  • Flux limits: The recovered plasma evolution model depends on the ratio of the local time step ∆t to the relaxation parameter τ.This scale dependence allows the discretized model to change with modeling scales.
  • Source-term limits: Implicit electromagnetic source treatment makes the time step unrestricted by the cyclotron period and confines particle motion to magnetic-field lines as time increases.The source-term linear system acts as a contractive projection for helical particle motion.
  • Asymptotic preservation: As rLi and λ̂D approach zero, the BGK-Maxwell asymptotic limits preserve synchronous electron-ion motion and quasi-neutrality.The summed equations converge to the MHD momentum equation.
  • Asymptotic preservation: Decreasing rLi yields a smooth transition from Ampere’s law to ideal Ohm’s law for electric-field calculation.In the continuum MHD regime, this removes the light-speed constraint on the time step.

5. Numerical experiments

UGKS is tested from the collisionless Vlasov regime through MHD, with numerical results matching theoretical kinetic behavior and recovering nonlinear damping effects.

  • MHD regime: The Brio-Wu shock-tube solution transitions from Euler behavior to Hall-MHD and ultimately ideal-MHD behavior as normalized Larmor radius decreases.The results compare density, velocity, and magnetic-field profiles across different normalized Larmor radii.
  • Linear Landau damping: Low-velocity particles absorb energy from the electric wave during Landau damping.The velocity distribution at x = 0 provides the corresponding particle-level diagnostic.
  • Nonlinear Landau damping: UGKS reproduces nonlinear Landau damping rates γ1 = −0.287 and γ2 = 0.078, consistent with published reference values.The velocity-distribution profiles show the nonlinear effect clearly.

5.3. Linear two stream instability

UGKS reproduces the theoretical growth rate of linear two-stream instability and resolves increasingly detailed nonlinear distribution structures as the mesh is refined.

  • Linear two-stream instability: The linear two-stream instability uses α = 0.001 and k = 0.2 with a fixed ion background balancing electron charge.The electric-field evolution is measured in the L2 norm.
  • Linear two-stream instability: UGKS predicts the same electric-field growth rate as linear theory for the two-stream instability.The growth rate is obtained from the time evolution of the electric field in the L2 norm.
  • Nonlinear two-stream instability: For nonlinear two-stream instability, detailed distribution-function structures require a very fine mesh with the second-order scheme.Velocity-distribution contours are compared at t = 70 for different mesh sizes.
  • Brio-Wu shock tube: The Brio-Wu test compares averaged density, velocity, ion-electron quantities, and magnetic-field profiles across normalized Larmor radii.The solutions progress from Euler behavior at large radius toward coupled-fluid Hall-MHD and ideal-MHD behavior at small radius.

5.6. Multiple scale shock tube problem

The multiple-scale shock-tube calculation follows UGKS from a collisionless Vlasov solution at fine resolution to an MHD solution at coarse resolution.

  • Kinetic regime: At the collisionless limit, UGKS approaches the Vlasov solution rather than the hydrodynamic two-fluid solution.This behavior occurs with Δx = 2 × 10^-4 rLi and a time step tied to the ion relaxation and plasma-frequency scales.
  • Transition regime: In the transition regime, UGKS, Vlasov, and hydrodynamic two-fluid solutions deviate, while UGKS is identified as physically reliable.The transition is examined using cell sizes and time steps that vary with the modeled scale.
  • Hydrodynamic regime: At t = 0.1, UGKS and the two-fluid system converge to the MHD solution in the hydrodynamic regime.The reported computation time for the UGKS solution is 126 seconds on a 3.40 GHz 4-core CPU.

5.7. Orszag-Tang Vortex

UGKS captures Orszag-Tang turbulence and resolves Larmor-radius-dependent magnetic reconnection, including transitions from Sweet-Parker diffusion to Hall-dominated structures.

  • Magnetic reconnection: Magnetic reconnection near the domain center merges two magnetic rings into one as time evolves.The reconnection mechanism changes with normalized Larmor radius.
  • Magnetic reconnection: At rLi = 0, reconnection has a large-aspect-ratio double-Y layer driven by magnetic diffusion in the Sweet-Parker mechanism.This is the ideal-MHD limit used as one reference case.
  • Magnetic reconnection: At rLi = 2.0, the Hall effect dominates and an X-point geometry is observed.The reconnection rate increases with higher energy-transfer efficiency as rLi increases.

5.8. Magnetic Reconnection

The UGKS is applied to magnetic reconnection in the transition regime, resolving reconnection behavior across varying normalized Larmor radii and plasma conditions. Its results include flux evolution, field topology, and near-conservation of total energy.

  • Pressure distributions: At rL = 0, the pressure distribution is compared with an ideal-MHD solution; at rL = 1.0, only the UGKS pressure distribution is shown.The comparison is made along y = 0.625π.
  • Magnetic-field topology: Magnetic reconnection occurs near the computational-domain center, with electron current sheets appearing as the normalized Larmor radius increases.The topology is shown for rLi = 0, 0.5, 1.0, and 2.0; current sheets are identified for rLi = 0.5, 1.0, and 2.0.
  • Reconnected flux: The UGKS reconnected flux is compared with other GEM simulations and predicts a fast reconnection rate.The reconnected flux is defined as the spatial average of |By(x, 0, t)| along the x direction.
  • Regime dependence: For rL = 0.5, the UGKS reconnected flux behaves like a resistive-MHD result, whereas for rL = 2 it approaches a Hall-MHD solution.The reported behavior changes with the normalized Larmor radius and plasma conditions.
  • Energy evolution: The total electromagnetic-plus-flow energy almost remains constant during the reconnection process.The simulation uses the GEM challenge problem's initial conditions and reports densities and momentum distributions at t = 40ω^-1.

6. Conclusion

The conclusion presents UGKS as a direct-modeling, multi-scale method for multi-species plasma transport. It recovers kinetic, two-fluid, and MHD behavior while addressing transitional-regime plasma dynamics and magnetic reconnection.

  • Method: UGKS constructs a multi-species plasma model from electron–ion collisions and coupling through the full Maxwell equations.The method describes evolution on mesh-size and time-step scales, producing multi-scale governing equations.
  • Multi-scale scope: The scheme covers flow physics beyond the isolated collisionless Vlasov and hydrodynamic MHD limits.Its stated scope includes reliable physical solutions in transitional regimes.
  • Regime recovery: UGKS recovers results from the kinetic Vlasov equation to the hydrodynamic two-fluid system as the time scale varies.The generalized Brio-Wu test demonstrates a smooth transition from neutral-fluid results to MHD solutions.
  • Validation and applications: The Orszag-Tang study shows that UGKS captures different MHD solutions and recovers magnetic reconnection under different conditions.The conclusion also points to fast magnetic reconnection at the Debye-length scale in the GEM case.
  • Broader applicability: The direct-modeling methodology is presented as applicable to multi-scale transport in rarefied gas dynamics, radiative transfer, and plasma physics.The conclusion identifies this as a broader methodological use of the UGKS framework.

Appendix: eigen-system of perfectly hyperbolic Maxwell equations

The appendix specifies the eigen-systems used for the perfectly hyperbolic Maxwell equations, including eigenvalues and left and right eigenvectors for the directional matrices.

  • Eigenvector matrices: The appendix presents the eigenvectors through displayed right- and left-eigenvector matrices for both directional systems.The matrices are included as part of the eigen-system construction for the hyperbolic Maxwell formulation.
  • A1 eigen-system: For A1, the eigenvalues are {c, c, cχ, cν, −c, −c, −cχ, −cν}.The right eigenvectors are given by the columns of a matrix, while the left eigenvectors are given by the rows of another matrix.
  • A2 eigen-system: For A2, the eigenvalues are {c, c, cχ, cν, −c, −c, −cχ, −cν}.The right eigenvectors are given by matrix columns, and the left eigenvectors by matrix rows.
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