Physics & Discretizations¶
The Vlasov–Poisson System¶
At its core, SPECTRAX-GK solves the Vlasov–Poisson system in 1D (space)–1V (velocity) for each plasma species \(s\).
The Vlasov equation describes phase-space evolution of the distribution function \(f_s(x,v,t)\):
Coupled with Poisson’s equation for the electric field:
Physical analogy¶
- Think of \(f_s(x,v,t)\) as a “cloud” of particles in phase space.
- The Vlasov equation:
particles move in straight lines (free streaming) but are accelerated by electric fields. - The Poisson equation:
the field is self-consistently generated by charge imbalances in that same particle cloud.
This feedback loop drives collective plasma effects: Landau damping, two-stream instability, bump-on-tail instability.
Fourier–Hermite Discretization¶
We expand in Fourier modes (for spatial variation) and Hermite polynomials (for velocity space):
- Fourier in \(x\): captures periodic structures like plasma waves.
- Hermite in \(v\): efficiently resolves velocity space fine structure.
Operators¶
The electric field couples through the zeroth Hermite coefficient:
The streaming operator (free particle motion) is tridiagonal in Hermite index \(n\), while the electric field couples Hermite modes across species.
Nonlinearity is handled with a pseudo-spectral method in real space:
with 2/3-rule dealiasing to maintain spectral accuracy.
Discontinuous Galerkin (DG)–Hermite¶
Instead of Fourier in \(x\), we can discretize spatial dependence with discontinuous Galerkin (DG):
- Local polynomial expansions in each spatial cell.
- Upwind fluxes handle advection \(v \partial_x f\).
- Periodic BCs by default.
In velocity space we still use Hermite polynomials.
The Poisson operator becomes a matrix \(P\) such that:
where \(\rho\) is the charge density projection.
This hybrid DG–Hermite scheme is more robust in strongly nonlinear regimes.
Energetics & Diagnostics¶
We track kinetic and field energy to ensure conservation and diagnose dynamics.
Kinetic energy of species s¶
Field energy¶
Example Phenomena¶
- Landau damping:
Initial perturbation damps due to phase mixing, without collisions. - Two-stream instability:
Counter-streaming electron beams excite growing plasma waves. - Bump-on-tail:
A superthermal tail destabilizes Langmuir waves.
Each of these is reproducible with SPECTRAX-GK by changing the input .toml.
References¶
- Landau, L. D. On the vibration of the electronic plasma. J. Phys. USSR, 1946.
- Cheng, C. Z., Knorr, G. Integration of the Vlasov Equation in Configuration Space. JCP, 1976.
- Boyd, J. P. Chebyshev and Fourier Spectral Methods. Dover, 2001.
- Shu, C.-W. Discontinuous Galerkin Methods. Springer, 2016.