1. Algorithm Development for the Full Two-Fluid Plasma System
University of Washington Department of Aeronautics & Astronautics
John Loverich
Ammar Hakim
Uri Shumlak

2. Overview Motivation Full Two-Fluid Model Preserving Divergence
Potential formulation
Auxiliary variables
Discontinuous Galerkin Method
Collisionless Reconnection

3. Motivation MHD is invalid in many plasma regimes
Microinstabilities and anomalous transport
Lower Hybrid Drift instability
Modified Two-Stream instability
Electron Kelvin Helmholtz instability
Weibel instability …
Two-fluid stability - FRC, z-pinch
Collisionless reconnection
Finite volume methods and discontinuous Galerkin methods have been used extensively in fluid mechanics. We would like to apply the same methods to Maxwell’s equations for the purpose of simpler algorithm design.

4. Full Two-Fluid Plasma Model: 5 Moment Fluid Equations
Species Continuity
Species Momentum
Species Energy
There are two fluids, electron fluid and ion fluid, each with complete inviscid fluid Equations + Lorentz force source terms. Higher moments of the Vlasov (collisionless Boltzmann) equation can be taken to improve the plasma model.

5. Full Two-Fluid Plasma Model: Maxwell’s Equations
Ampere’s Law
Faraday’s Law
Poisson’s Equation
Magnetic Flux
The fluids are coupled to each other through the electromagnetic fields.

6. Maxwell’s Equations – Mixed Potential Formulation
The finite volume method absolutely required divergence cleaning in order to get proper solution to problems with in plane magnetic fields.
The potential equations can be used to ensure the divergence equations are satisfied.
The potential equations are re-written as 16 first order equations so that Riemann solvers can be applied.
The Lorentz gauge condition must still be satisfied. Errors in this constraint remain small.

7. Perfectly Hyperbolic Maxwell’s Equations
Another approach to dealing with the divergence conditions is to use the perfectly hyperbolic Maxwell’s equations
Auxiliary variables are used to propagate errors in the solution out of the domain at some pre-determined speed.
We have not yet noticed any unphysical effects produced by the auxiliary variables.

8. Why use the discontinuous Galerkin method?
Source terms
included naturally
Temporal accuracy
long time integration
Spatial accuracy
High order methods are good at balancing sources and fluxes near equilibrium (this is very important in two-fluid equations)
Divergence
Finite volume methods require divergence cleaning to gain solutions to problems with in plane magnetic fields
Explicit
easy to parallelize
Efficiency
Higher order methods can be computationally more efficient …

9. Discontinuous Galerkin Method
Constant term
Linear variation
Solution does not need to be continuous at cell edges

10. Discontinuous Galerkin Method
Start with a general balance law,
The Q are represented as a linear combination of basis functions
Multiply the balance law by the same set of basis functions and integrate over a volume element,

11. Discontinuous Galerkin Method
Move the derivative off the flux F and onto the basis functions using integration by parts,
In regular geometries with orthogonal basis function the equation becomes.

12. Discontinuous Galerkin Method
We still have a few things to evaluate.
Surface Fluxes – Approximate Riemann Flux
Integrals – Gaussian Quadrature
Time Derivatives – Runge-Kutta methods
Extension to general geometries is very easy!
Calculate Jacobians at each quadrature point
Calculate basis function gradients in global coordinates
Calculate a local mass matrix

13. Collisionless Reconnection
Image borrowed from Journal of Geophysical Research, Vol. 106, No. A3, Pg , March 1, 2001

14. Collisionless Reconnection
The following simulation is based off a widely explored collisionless magnetic reconnection problem called the GEM challenge.

15. Collisionless Reconnection
After 25/Wci the 2nd order solution differs substantially from the 3rd order due to the formation of a large magnetic island in the 2nd order solution.
Total electron current at T=25/Wci

16. Collisionless Reconnection
At a resolution of 512X256 the 2nd and 3rd order methods are essentially the same. The 3rd order method achieves a correct solution at lower grid resolution.
Total electron current at T=25/Wci

17. Collisionless Reconnection
Comparison of reconnected magnetic flux for the full two-fluid solution using 3rd order discontinuous Galerkin method against solutions published by M. Shay, Journal of Geophysical Research, Vol. 106 No A3, Pg

18. Conclusion
We are interested in the two-fluid plasma model because MHD is inadequate.
A discontinuous Galerkin method for the two-fluid plasma system has been described.
Techniques that preserve divergence have been successfully applied to the two-fluid system.
The algorithm produces results in agreement with other techniques.