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:
http://python-hydro.github.io/pyro2/
- Getting started
- Core Data Structures
- Solvers
- Working with data
- Understanding the algorithms
- Regression and unit testing
- python 2.7
- Acknowledgements
- Getting help
-
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
,f2py
,matplotlib
, andh5py
installed. On a Fedora system, this can be accomplished by doing:dnf install python3-numpy python3-numpy-f2py python3-matplotlib python3-matplotlib-tk python3-h5py
(note, for older Fedora releases, replace
dnf
withyum
. For python 2.x, leave off the2
in the package names.) -
You also need to make sure gfortran is present on you system. On a Fedora system, it can be installed as:
dnf install gcc-gfortran
-
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 thepyro2/
directory. -
Define the environment variable
PYRO_HOME
to point to thepyro2/
directory (only needed to regression testing) -
Build the Fortran source. In
pyro2/
type./mk.sh
-
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.
-
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
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_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. -
incompressible
: a second-order cell-centered approximate projection method for the incompressible equations of hydrodynamics. -
diffusion
: a Crank-Nicolson time-discretized solver for the constant-coefficient diffusion equation. -
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). -
swe
: a solver for the shallow water equations.
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 fromcylindrical-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 fromsod-exact.out
).usage:
./sod_compare.py file
-
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
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
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
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.
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