1D Convection-Diffusion Solver

This repository contains a Python implementation of a 1D finite-difference solver for the convection–diffusion equation.

$$\frac{d}{dx} \left( u \phi \right) = \frac{d}{dx} \left( \Gamma \frac{d\phi}{dx} \right)$$

Where:

• $$( \phi(x) )$$: scalar field (e.g., temperature, concentration)
• $$( u )$$: velocity (assumed constant)
• $$( \Gamma )$$: diffusion coefficient (can be constant or variable)

It includes multiple discretization schemes and supports deferred correction to improve accuracy.

Features

• ✅ Multiple convection schemes:
  • First-order Upwind (UD1)
  • Central Difference (CD2)
  • Third-order Upwind (UD3)
  • Fourth-order Central Difference (CD4)
• ✅ Diffusion discretization:
  • Second-order Central Difference (CD2)
  • Fourth-order Central Difference (CD4)
• ✅ High-order deferred correction for both terms
• ✅ Efficient tridiagonal solver (TDMA)
• ✅ Automated convergence study & log-log error plot
• ✅ Output table with observed convergence rates

Project Structure

. ├── fd1d.py # Main solver and convergence logic ├── requirements.txt # Python dependencies ├── ConvergenceRates.png ├── convergence_table.csv ├── .gitignore └── README.md

Getting Started

Create a virtual environment

python3 -m venv venv
source venv/bin/activate          #On Windows: venv\Scripts\activate
pip install -r requirements.txt
python ./fd1d.py 

This will:

• solve the convection–diffusion equation

• generate convergenceRates.png

• generate convergence_table.csv

Sample Results

The solver estimates the order of convergence using least squares on log-log data. Example:

Scheme Observed Order

UD1 / CD2 ≈ 1 CD2 / CD2 ≈ 2 UD3 / CD2 ≈ 3 CD4 / CD2 ≈ 4 CD4 / CD4 ≈ 4

Dependencies

numpy
matplotlib
pandas

Notes

This solver is designed for testing and educational analysis of finite-difference discretizations of the convection–diffusion equation in 1D. This is inspired by Peric's Lectures on CFD.

Feel free to extend it to include:

• Variable coefficients

• 2D problems

• Time-dependent solutions

Contributions

PRs and improvements are welcome. If you find a bug or want to request a feature, please open an issue.