baron-sakender 0.0.1

2D Ideal MHD Simulator using HLL Riemann Solver with Dedner Divergence Cleaning

baron-sakender: JAX-Accelerated 2D Ideal MHD Solver

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A finite volume solver for the two-dimensional ideal magnetohydrodynamic (MHD) equations with hyperbolic-parabolic divergence cleaning. This implementation employs the HLL approximate Riemann solver and second-order Runge-Kutta time integration, with optional GPU acceleration via JAX.

Governing Equations

The ideal MHD system describes the dynamics of electrically conducting fluids permeated by magnetic fields. In conservative form with the Dedner divergence cleaning formulation, the equations read:

$$ \partial_t \mathbf{U} + \partial_i \mathbf{F}^i = \mathbf{S} $$

where $\partial_t \equiv \partial/\partial t$ denotes the temporal derivative, $\partial_i \equiv \partial/\partial x^i$ the spatial derivative with respect to coordinate $x^i$, $\mathbf{U}$ the state vector, $\mathbf{F}^i$ the flux tensor, and $\mathbf{S}$ the source term.

State Vector

The conservative state vector $\mathbf{U} \in \mathbb{R}^7$ is defined as:

$$ \mathbf{U} = \begin{pmatrix} \rho \ \rho v^x \ \rho v^y \ B^x \ B^y \ E \ \psi \end{pmatrix} $$

Conservation Laws

Mass conservation:

$$ \partial_t \rho + \partial_i (\rho v^i) = 0 $$

Momentum conservation:

$$ \partial_t (\rho v^j) + \partial_i \left( \rho v^i v^j + P^* \delta^{ij} - B^i B^j \right) = 0 $$

where $\delta^{ij}$ is the Kronecker delta and $P^*$ denotes the total pressure:

$$ P^* = p + \frac{1}{2} B^k B_k $$

Magnetic induction with divergence cleaning:

$$ \partial_t B^j + \partial_i \left( v^i B^j - B^i v^j \right) + \partial^j \psi = 0 $$

Energy conservation:

$$ \partial_t E + \partial_i \left[ (E + P^*) v^i - B^i (v_k B^k) \right] = 0 $$

Divergence cleaning (Dedner et al. 2002):

$$ \partial_t \psi + c_h^2 , \partial_i B^i = -\frac{c_h^2}{c_p^2} \psi $$

Closure Relations

Total energy density:

$$ E = \frac{1}{2} \rho v^i v_i + \frac{p}{\gamma - 1} + \frac{1}{2} B^i B_i $$

Equation of state (ideal gas):

$$ p = (\gamma - 1) \left( E - \frac{1}{2} \rho v^i v_i - \frac{1}{2} B^i B_i \right) $$

Characteristic Speeds

Sound speed:

$$ c_s = \sqrt{\frac{\gamma p}{\rho}} $$

Alfvén speed:

$$ v_A = \frac{|B|}{\sqrt{\rho}} = \sqrt{\frac{B^i B_i}{\rho}} $$

Fast magnetosonic speed:

$$ c_f = \sqrt{c_s^2 + v_A^2} $$

Variable Definitions

Symbol	Description	SI Unit
$\rho$	Mass density	[kg m⁻³]
$v^i$	Velocity component	[m s⁻¹]
$B^i$	Magnetic field component	[T]
$p$	Thermal pressure	[Pa]
$P^*$	Total pressure (thermal + magnetic)	[Pa]
$E$	Total energy density	[J m⁻³]
$\psi$	Divergence cleaning potential	[T m s⁻¹]
$\gamma$	Adiabatic index	[—]
$c_s$	Sound speed	[m s⁻¹]
$v_A$	Alfvén speed	[m s⁻¹]
$c_f$	Fast magnetosonic speed	[m s⁻¹]
$c_h$	Hyperbolic cleaning speed	[m s⁻¹]
$c_p$	Parabolic damping rate	[s⁻¹]

Normalization (Code Units)

The code employs dimensionless variables normalized by reference quantities:

Reference	Symbol	Description
$\rho_0$	Reference density	[kg m⁻³]
$B_0$	Reference magnetic field	[T]
$L_0$	Reference length	[m]
$v_A = B_0 / \sqrt{\mu_0 \rho_0}$	Alfvén velocity	[m s⁻¹]
$\tau_A = L_0 / v_A$	Alfvén time	[s]
$P_0 = B_0^2 / \mu_0$	Magnetic pressure	[Pa]

where $\mu_0 = 4\pi \times 10^{-7}$ [H m⁻¹] is the vacuum permeability.

Numerical Methods

• Spatial discretization: Finite volume method on uniform Cartesian grid
• Riemann solver: HLL (Harten-Lax-van Leer) approximate solver
• Time integration: RK2 (Heun's method), second-order accurate
• Divergence cleaning: Hyperbolic-parabolic formulation (Dedner et al. 2002)
• Boundary conditions: Periodic
• Positivity preservation: Density and pressure floors

Installation

Requirements

• Python 3.9 or later
• NumPy < 2.0 (for compatibility with compiled dependencies)

From Source

git clone https://github.com/sandyherho/baron-sakender.git
cd baron-sakender
pip install -e ".[dev,netcdf]"

Dependencies

Core:

• NumPy — Array operations
• SciPy — Scientific computing
• JAX — Accelerated numerical computing
• Matplotlib — Visualization
• pandas — Data handling
• tqdm — Progress bars

Optional:

• netCDF4 — CF-compliant output
• pytest — Testing framework
• black — Code formatting

Usage

Command Line Interface

# Run Orszag-Tang vortex (standard benchmark)
baron-sakender case1

# Run all test cases
baron-sakender --all

# Run with GPU acceleration
baron-sakender case1 --gpu

# Run from configuration file
baron-sakender -c configs/case1_orszag_tang.txt

Python API

from baron_sakender import MHDSystem, MHDSolver

# Initialize system
system = MHDSystem(nx=256, ny=256, gamma=5/3)
system.init_orszag_tang()

# Create solver and run
solver = MHDSolver(cfl=0.4, use_gpu=False)
result = solver.run_with_diagnostics(system, t_end=3.0, save_dt=0.1)

# Access results
times = result['times']
snapshots = result['snapshots']
conservation = result['conservation_history']

Available Test Cases

Case	Description	Configuration
case1	Orszag-Tang vortex	Standard MHD turbulence benchmark
case2	Strong magnetic field	Low plasma beta ($\beta = 0.1$) regime
case3	Current sheet	Harris-like configuration
case4	Alfvén wave	Wave propagation test

Output Formats

• PNG: Static field plots (density, magnetic pressure, current, vorticity)
• GIF: Animated density evolution
• CSV: Time series of conservation and stability metrics
• NetCDF: CF-1.8 compliant data with full metadata

Testing

# Run all tests
pytest tests/ -v

# Run with coverage
pytest tests/ -v --cov=baron_sakender --cov-report=html

Project Structure

baron-sakender/
├── src/baron_sakender/
│   ├── core/
│   │   ├── mhd_system.py    # Physical system definition
│   │   ├── integrator.py    # JAX-accelerated solver
│   │   └── metrics.py       # Diagnostic computations
│   ├── io/
│   │   ├── config_manager.py
│   │   └── data_handler.py
│   ├── visualization/
│   │   └── animator.py
│   └── cli.py               # Command line interface
├── configs/                  # Configuration files
├── tests/                    # Test suite
└── pyproject.toml

References

• Dedner, A., Kemm, F., Kröner, D., Munz, C.-D., Schnitzer, T., & Wesenberg, M. (2002). Hyperbolic divergence cleaning for the MHD equations. Journal of Computational Physics, 175(2), 645–673.
• Harten, A., Lax, P. D., & van Leer, B. (1983). On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review, 25(1), 35–61.
• Orszag, S. A., & Tang, C.-M. (1979). Small-scale structure of two-dimensional magnetohydrodynamic turbulence. Journal of Fluid Mechanics, 90(1), 129–143.

License

MIT License. See LICENSE for details.

Authors

Sandy H. S. Herho, Nurjanna J. Trilaksono

Citation

If you use this software in your research, please cite:

@software{baron_sakender,
  author = {Herho, Sandy H. S. and Trilaksono, Nurjanna J.},
  title = {baron-sakender: JAX-Accelerated 2D Ideal MHD Solver},
  year = {2025},
  url = {https://github.com/sandyherho/baron-sakender}
}