Spectral NS Solver 1

Overview

This library numerically solves the incompressible Navier-Stokes equations with a scalar field in two- and three-dimensional Cartesian domains using the spectral method. It was developed as a self-study project to explore the differences between finite-difference methods and spectral methods.

Features

• Fourier-Galerkin method
• Orszag–Patterson algorithm
• 2/3 de-aliasing
• Pencil-based MPI parallelization for scaling up to 10⁴ processes
• Fourth-order Runge-Kutta method (for nonlinear terms) combined with the integrating-factor technique (for linear terms) for temporal integration

Refer to the documentation for details (currently under construction).

Dependencies

• C compiler
• GNU Make
• MPI
• FFTW3

To easily initialize the flow field, it is recommended to use:

• Python with NumPy (see initial_condition/main.py)

Quick Start

1. Set up your workspace

   mkdir -p /path/to/your/directory
   cd /path/to/your/directory

2. Clone the repository

   git clone --recurse-submodules https://github.com/NaokiHori/SpectralNSSolver1
   cd SpectralNSSolver1

3. Set the initial condition

   The velocity field must be solenoidal, while the scalar field can be arbitrary. main.py provides several example initial conditions.

   cd initial_condition
   python3 main.py 0
   cd ..

4. Build the solver

   make clean
   make output
   make all

5. Run the simulation

   Execution parameters are defined in exec.sh.

   bash exec.sh

   The runtime depends on your system specifications.

6. Output and Visualization

   The flow fields are stored in output/save/ as NPY files. These velocities are in the spectral domain, so an inverse Fourier transform (with normalization) is needed to obtain physical velocities.

   If the necessary Python libraries are installed, you can visualize the results with:

   python3 visualise/2d.py

3D Simulation

To run a 3D simulation, switch to the 3d branch and recompile all source files. You will also need to regenerate the initial flow field.

Reference

• Canuto et al., Spectral Methods - Fundamentals in Single Domains, Springer
• Canuto et al., Spectral Methods - Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer

Acknowledgement

I would like to thank Dr. Chris Howland for valuable discussions.