README.md

1D Bi-Plasma Simulation

A C++ simulation code for modeling 1D fluid plasma dynamics with coupled ion and electron perturbations using finite difference methods.

Description

This project simulates the temporal evolution of a one-dimensional bi-species plasma system. It solves the coupled fluid equations for ion and electron density/velocity perturbations along with the electric field. The goal is to study wave propagation and instabilities in a plasma, keeping the second order terms.

Physical Model

The code solves the following system of equations:

1. Continuity equation (for species s = i, e):

$$ \frac{\partial n_{s1}}{\partial t} + (n_{s0} + n_{s1}) \frac{\partial u_{s1}}{\partial x} + u_{s1} \frac{\partial n_{s1}}{\partial x} = 0 $$

2. Momentum equation (for species s = i, e):

$$ \frac{\partial u_{s1}}{\partial t} + u_{s1} \frac{\partial u_{s1}}{\partial x} = \frac{q_s E}{m_s} - \frac{1}{m_s n_s}\frac{\partial P_{s}}{\partial x} $$

With the pressure gradient:

$$ \frac{1}{n_{s}} \frac{\partial P_{s}}{\partial x} = \frac{P_{s0}}{n_{s0}} \left[\frac{\gamma_s}{n_{s0}}\frac{\partial n_{s1}}{\partial x}\left(1 + \frac{\gamma_s n_{s1}}{n_{s0}}\right) - 2\frac{\gamma_s n_{s1}}{n_{s0}^2} \frac{\partial n_{s1}}{\partial x}\right] $$

3. Poisson's equation:

$$ \frac{\partial E}{\partial x} = \frac{1}{\varepsilon_0} (q_i n_{i1} + q_e n_{e1}) $$

Where:

• n_s1: Density perturbation [m⁻³]
• u_s1: Velocity perturbation [m/s]
• E: Electric field [V/m]
• P_s: Pressure [Pa]
• q_s: Charge [C]
• m_s: Mass [kg]

Numerical Methods

The solver implements a 2nd-order 1D finite volume scheme for nonlinear electrostatic plasma dynamics:

• Spatial discretization: Conservative form with MUSCL reconstruction (minmod limiter) for 2nd-order accuracy while avoiding non-physical oscillations
• Numerical fluxes: Rusanov scheme (Local Lax–Friedrichs) at interfaces, ensuring robustness and stability
• Time integration: MacCormack-type predictor–corrector scheme (2nd-order), with predictor step (advection + sources) and corrector step (backward reconstruction + time averaging)
• Poisson's equation: Tridiagonal linear solver on centered finite differences, solved at each substep for self-consistent electric field evolution
• Boundary conditions: Periodic
• CFL condition:

$$ \Delta t \leq \text{cfl} \times \frac{\Delta x}{C_s} $$

where $C_s$ is the characteristic plasma sound speed.

Dependencies

• C++ compiler with C++11 support (g++, clang++)
• Eigen3 library (included in include/Eigen/)
• CMake (version 3.10 or higher)
• Python 3 with matplotlib and numpy (for visualization)

Project Structure

.
├── CMakeLists.txt          # CMake configuration file
├── README.md               # This file
├── include/
│   ├── constants.hpp       # Physical constants
│   ├── plasma.hpp          # PlasmaSystem class definition
│   └── Eigen/              # Eigen library headers
├── src/
│   ├── main.cpp            # Main simulation program
    ├── plasmas.cpp         # PlasmaSystem class implementation
│   └── plot_time.py        # Python visualization script
├── build/                  # Build directory
├── data/                   # Output data files
└── image/                  # Generated plots

Building the Project

1. Create and navigate to the build directory:

   mkdir -p build
   cd build

2. Configure with CMake:

   cmake ..

3. Compile:

   make

This will generate the bi_plasma executable in the build/ directory.

Running the Simulation

From the build/ directory:

./bi_plasma

The program will:

1. Initialize the plasma system with a Gaussian density perturbation
2. Run the time evolution simulation
3. Save data to data/ directory
4. Automatically run the visualization script to generate plots

Output Files

The simulation generates the following data files in the data/ directory:

• time.txt: Time array [s]
• x_grid.txt: Spatial grid [m]
• n_i1_time.txt: Ion density perturbation evolution
• u_i1_time.txt: Ion velocity perturbation evolution
• n_e1_time.txt: Electron density perturbation evolution
• u_e1_time.txt: Electron velocity perturbation evolution
• E_time.txt: Electric field evolution

Generated plots are saved in the image/ directory.

Exemple of results

License

This project is licensed under the MIT License - see the LICENSE file for details.

Authors

Ewan Bataille