vlax1d 0.1.1

A fast 1D Vlasov Poisson solver using JAX

One Dimension Vlasov-Poisson Equation JAX Solver

This repository provides a numerical solver for the one-dimensional Vlasov-Poisson equation using JAX for high-performance computing. The solver leverages GPU acceleration and JAX's automatic differentiation to efficiently simulate plasma wave dynamics, including Landau damping and instability phenomena.

Project Overview

(add some explanation of calculation) This solver numerically integrates the 1D linear Vlasov-Poisson equation to analyze wave propagation, damping rates, and stability in plasma physics. The solver utilizes JAX to achieve high-performance computations, supporting automatic differentiation and GPU/TPU acceleration.

Features

• High-performance JAX-based solver: Leverages JAX's automatic differentiation and just-in-time (JIT) compilation.
• Robust analysis tools: Peak detection, linear fitting, and data export to CSV.
• Customizable parameters: Easy-to-adjust parameters such as wave number (k), collision frequency (nu), and the mode number for all species (m_max).

Installation

pip install Oned-Vlasov-Poisson-jax-solver

Usage

To run simulations and analyze damping rates:

python two_stream_instability.py

Adjust parameters within the script:

# Set the maximum number of all modes (the dimension of the state variable)
# Which should be an even number because the subsequent code will divide it into two groups
m_max = 40

# Define the time range of the simulation
# From t = 0 to t = 200, a total of 2001 time points (i.e. Δt = 0.1)
t_span = jnp.linspace(0, 200, 2001)

# Set of wave numbers (k values) to loop over or analyze.
# These represent different spatial scales of perturbations in the system.
k_values = jnp.linspace(0.01, 0.40, 40)

# Collision frequency (nu), representing the strength of the damping term.
# Higher values introduce stronger diffusion (e.g., Landau damping, viscosity).
nu = 2

# Average velocities for two interacting populations (e.g., beam and background).
# Used to model counter-propagating or asymmetric systems.
v_e = jnp.array([1, -1])

# Charge weights or coefficients associated with the populations.
# For population one and two.
q = jnp.array([1, 1])

Mathematical Background

We start with the 1D Vlasov equation:

∂f/∂t + v ∂f/∂x + (qE/m) ∂f/∂v = 0

After linearizing and Fourier-transforming:

∂(δf)/∂t + i·k·v·(δf) + (q·δE/m) ∂fₘ/∂v = 0

We expand δf(v, t) in Hermite polynomials:

δf(v, t) = Σₙ fₙ(t) Hₙ((v - vₑ)/α)

Then, projecting onto each Hermite mode:

∂fₙ/∂t = -i√[(n+1)/2] fₙ₊₁ - i√[n/2] fₙ₋₁ - i·vₑ·fₙ - C·f₀·δₙ₁

where C ∝ 1/k from Poisson's equation.

for more detail please check plasma_equation in docs

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Project Structure

OneD_Vlasov_Poisson_JAX_Solver/
├── .github/
│   └── workflows/
│       └── test.yml              # GitHub Actions for CI
│
├── data/
│   ├── damping_rate.csv          # Reference damping data
│   └── growth_rate.csv           # Reference instability growth data
│
├── dist/                         # Python package builds
│   ├── oned_vlasov_poisson_jax_solver-*.tar.gz
│   └── oned_vlasov_poisson_jax_solver-*.whl
│
├── docs/
│   ├── plasma_equation.pdf       # Mathematical background
│   ├── requirements.txt          # Build dependencies for Sphinx
│   └── VLAX1D_logo.png            # Project logo
│
├── examples/
│   ├── Landau_damping.py         # Example: Landau damping
│   └── two_stream_instability.py # Example: Two-stream instability
│
├── source/
│   ├── conf.py                   # Sphinx configuration
│   ├── index.rst                 # Sphinx main index
│   ├── installation.rst
│   ├── modules.rst
│   └── usage.rst
│
├── src/
│   ├── Oned_Vlasov_Poisson_jax_solver.egg-info/ # Package metadata
│   ├── __init__.py
│   ├── equations.py               # Equation definitions
│   ├── plot.py                    # Plotting functions
│   ├── Refactoring.py             # Hermite reconstruction and utilities
│   └── solver.py                  # Solver using Diffrax
│
├── tests/
│   ├── __init__.py
│   ├── test_equations.py
│   ├── test_plot.py
│   ├── test_Refactoring.py
│   └── test_solver.py
│
├── .coverage
├── coverage.xml
├── LICENSE
├── Makefile
├── make.bat
├── MANIFEST.in
├── pyproject.toml
├── pytest.ini
├── README.md                     # Project overview
├── readthedocs.yml                # ReadTheDocs configuration
├── requirements.txt              # Python runtime requirements
└── setup.py                       # Setup script for pip installation

Contributing

Contributions are welcome! Please open an issue or submit a pull request.