A simple python-based tutorial on computational methods for hydrodynamics

pyro is a computational hydrodynamics code that presents two-dimensional solvers for advection, compressible hydrodynamics, diffusion, incompressible hydrodynamics, and multigrid, all in a finite-volume framework. The code is mainly written in python and is designed with simplicity in mind. The algorithms are written to encourage experimentation and allow for self-learning of these code methods.

The latest version of pyro is always available at:

https://github.com/python-hydro/pyro2

The project webpage, where you'll find documentation, plots, notes, etc. is here:

https://pyro2.readthedocs.io/

Table of Contents

• Getting started
• Core Data Structures
• Solvers
• Working with data
• Understanding the algorithms
• Regression and unit testing
• python 2.7
• Acknowledgements
• Getting help

Getting started

• By default, we assume python 3.4 or later. Instructions to run with python 2.7 are given below, but it is recommended you switch to python 3.x

• There are a few steps to take to get things running. You need to make sure you have numpy, numba, matplotlib, and h5py installed. On a Fedora system, this can be accomplished by doing:

  dnf install python3-numpy python3-numba python3-matplotlib python3-matplotlib-tk python3-h5py

  (note, for older Fedora releases, replace dnf with yum. For python 2.x, leave off the 3 in the package names.)

• Not all matplotlib backends allow for the interactive plotting as pyro is run. One that does is the TkAgg backend. This can be made the default by creating a file ~/.matplotlib/matplotlibrc with the content:

  backend: TkAgg

  You can check what backend is your current default in python via:

  import matplotlib.pyplot
  print matplotlib.pyplot.get_backend()

• If you want to run the unit tests, you need to have pytest installed.

• The remaining steps are:

  • Set the PYTHONPATH environment variable to point to the pyro2/ directory (note this is only needed if you wish to use pyro as a python module - this step is not necessary if you only run pyro via the commandline using the pyro.py script).

  • Define the environment variable PYRO_HOME to point to the pyro2/ directory (only needed for regression testing)

  • Run a quick test of the advection solver:

    ./pyro.py advection smooth inputs.smooth

    you should see a graphing window pop up with a smooth pulse advecting diagonally through the periodic domain.

Core Data Structures

The main data structures that describe the grid and the data the lives on the grid are described in a jupyter notebook:

https://github.com/python-hydro/pyro2/blob/master/mesh/mesh-examples.ipynb

Many of the methods here rely on multigrid. The multigrid solver is demonstrated in the juputer notebook:

https://github.com/python-hydro/pyro2/blob/master/multigrid/multigrid-examples.ipynb

Solvers

pyro provides the following solvers (all in 2-d):

• advection: a second-order unsplit linear advection solver. This uses characteristic tracing and corner coupling for the prediction of the interface states. This is the basic method to understand hydrodynamics.

• advection_fv4: a fourth-order accurate finite-volume advection solver that uses RK4 time integration.

• advection_nonuniform: a solver for advection with a non-uniform velocity field.

• advection_rk: a second-order unsplit solver for linear advection that uses Runge-Kutta integration instead of characteristic tracing.

• advection_weno: a method-of-lines WENO solver for linear advection.

• compressible: a second-order unsplit solver for the Euler equations of compressible hydrodynamics. This uses characteristic tracing and corner coupling for the prediction of the interface states and a 2-shock or HLLC approximate Riemann solver.

• compressible_fv4: a fourth-order accurate finite-volume compressible hydro solver that uses RK4 time integration. This is built from the method of McCourquodale and Colella (2011).

• compressible_rk: a second-order unsplit solver for Euler equations that uses Runge-Kutta integration instead of characteristic tracing.

• compressible_sdc: a fourth-order compressible solver, using spectral-deferred correction (SDC) for the time integration.

• diffusion: a Crank-Nicolson time-discretized solver for the constant-coefficient diffusion equation.

• incompressible: a second-order cell-centered approximate projection method for the incompressible equations of hydrodynamics.

• lm_atm: a solver for the equations of low Mach number hydrodynamics for atmospheric flows.

• lm_combustion: (in development) a solver for the equations of low Mach number hydrodynamics for smallscale combustion.

• multigrid: a cell-centered multigrid solver for a constant-coefficient Helmholtz equation, as well as a variable-coefficient Poisson equation (which inherits from the constant-coefficient solver).

• particles: a solver for Lagrangian tracer particles.

• swe: a solver for the shallow water equations.

Working with data

In addition to the main pyro program, there are many analysis tools that we describe here. Note: some problems write a report at the end of the simulation specifying the analysis routines that can be used with their data.

• compare.py: this takes two simulation output files as input and compares zone-by-zone for exact agreement. This is used as part of the regression testing.

  usage: ./compare.py file1 file2

• plot.py: this takes the an output file as input and plots the data using the solver's dovis method.

  usage: ./plot.py file

• analysis/

  • dam_compare.py: this takes an output file from the shallow water dam break problem and plots a slice through the domain together with the analytic solution (calculated in the script).

    usage: ./dam_compare.py file

  • gauss_diffusion_compare.py: this is for the diffusion solver's Gaussian diffusion problem. It takes a sequence of output files as arguments, computes the angle-average, and the plots the resulting points over the analytic solution for comparison with the exact result.

    usage: ./gauss_diffusion_compare.py file*

  • incomp_converge_error.py: this is for the incompressible solver's converge problem. This takes a single output file as input and compares the velocity field to the analytic solution, reporting the L2 norm of the error.

    usage: ./incomp_converge_error.py file

  • plotvar.py: this takes a single output file and a variable name and plots the data for that variable.

    usage: ./plotvar.py file variable

  • sedov_compare.py: this takes an output file from the compressible Sedov problem, computes the angle-average profile of the solution and plots it together with the analytic data (read in from cylindrical-sedov.out).

    usage: ./sedov_compare.py file

  • smooth_error.py: this takes an output file from the advection solver's smooth problem and compares to the analytic solution, outputting the L2 norm of the error.

    usage: ./smooth_error.py file

  • sod_compare.py: this takes an output file from the compressible Sod problem and plots a slice through the domain over the analytic solution (read in from sod-exact.out).

    usage: ./sod_compare.py file

Understanding the algorithms

There is a set of notes that describe the background and details of the algorithms that pyro implements:

http://bender.astro.sunysb.edu/hydro_by_example/CompHydroTutorial.pdf

The source for these notes is also available on github:

https://github.com/Open-Astrophysics-Bookshelf/numerical_exercises

Regression and unit testing

The test.py script will run several of the problems (as well as some stand-alone multigrid tests) and compare the solution to stored benchmarks (in each solver's tests/ subdirectory).

It will also invoke the python pytest module to run the unit tests for the different modules in pyro.

Tests are run nightly and reported here:

http://bender.astro.sunysb.edu/hydro_by_example/download/_stage/pyro2/tests.out

python 2.7

To run with python 2.7, you need to build the compiled code as:

PYTHON=python2 ./mk.sh

and then run explicitly giving the name of the python interpreter on the commandline as, for example:

python2 ./pyro.py compressible sedov inputs.sedov

Acknowledgements

If you use pyro in a class or workshop, please e-mail us to let us know (we'd like to start listing these on the website).

If pyro was used for a publication, please cite the article found in the CITATION file.

Getting help

Join the mailing list to say up-to-date:

https://groups.google.com/forum/#!forum/pyro-code