Space-time waveform relaxation for Navier-Stokes

This repository is a suppliment to the paper Space-time waveform relaxation for Navier-Stokes, and contains the implementation required to reproduce the numerical experiments presented within.

Installation

This repository depends on:

Firedrake

The Firedrake installation script can be downloaded here. The latest version can be installed with

bash python3 firedrake-install

The Firedrake version used in the associated paper can be installed with the command

bash python3 firedrake-install --doi 10.5281/zenodo.12657473

The Zenodo URL for Firedrake can be found here, and has DOI 10.5281/zenodo.12657473.

Pandas

Data is visualised (in *_vis.py) with pandas, which should be pip installed into the virtual environment with

bash pip3 install pandas==1.5.0

Code structure

This code solves 3D finite element discretisations, where two of the dimensions are spatial and one is temporal. Both the heat equation and imcompressible Navier-Stokes without external forcing are solved. The codebase is split into three test problems.

• heat*.py: Code used to solve the heat equation.

• chorin*.py: Code used to solve Navier-Stokes to approximate the exact solution

math \begin{equation} \begin{split} \vec{u}(t,\vec{x}) & = \begin{pmatrix} - \cos{\pi x_1} \sin{\pi x_2} \\ \sin{\pi x_1} \cos{\pi x_2} \end{pmatrix} e^{-2\pi^2 t} \\ \phi(t,\vec{x}) & = R \frac{\pi}{4} \left( \cos{2\pi x_1} + \cos{2\pi x_2}\right) e^{-4\pi^2 t} . \end{split} \end{equation}

• lid*.py: Code used to solve Navier-Stokes for lid-driven cavity test problem.

For more information on each test case please see the associated paper.

Function breakdown

For each test problem has multiple python files associated with it, which we now briefly outline the functionality of.

{problem}.py

Runs the associated problem with the parameters fixed by the parameters class at the top of the file. By default the WRMG solver is used, although by changing parameters.solver='lu' a direct solver may also be used. By default solution plots will be generated and saved to the plots/ subfolder and may be visualised in paraview.

{problem}_caller.py

Calls the associated problem with optional arguments which can be passed to the parameters class. These files are auxiliary but heavily relied on for benchmarking. Run python3 {problem}_caller.py --help to see available options here. Note a default run will use WRMG but direct solvers can be used by passing the flag --lu.

{problem}_generator.py

Generates the timing benchmarks displayed in the paper using 8 MPI cores (by default). Runs can also be computed in parallel (by increasing MaxProcesses parameters in script) but this is not recommended as computations can be memory intensive. To run timings for WRMG run

bash python3 {problem}_generator.py

and for the direct solver run

bash python3 {problem}_generator.py --flags 'lu'

Timings will be output in the terminal as well as saved to data/ subdirectory as a csv file.

{problem}_vis.py

A helper function to visualise computational timings both in the terminal and to save them as csv files.

lid_stepper.py

A variation of lid.py where the space-time finite element method is solved over a single time slab sequentually in a time-stepping manor. Can be called in lid_caller.py with the flag --stepper.

lid_parallel_generator.py

Generates parallelisation study for lid-driven cavity presented in the paper by varying the number of MPI cores for the WRMG solver. To generate the timings for the time-stepping approach pass the flag --flags 'stepper'.