Simple one-dimensional examples of various hydrodynamics techniques

This is a collection of simple python codes (+ a few Fortran ones) that demonstrate some basic techniques used in hydrodynamics codes. All the codes are standalone -- there are no interdependencies.

These codes go together with the lecture notes at:

https://open-astrophysics-bookshelf.github.io/numerical_exercises/

and with the pyro2 code:

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

• advection/

  • advection.py: a 1-d second-order linear advection solver with a wide range of limiters.

  • fdadvect_implicit.py: a 1-d first-order implicit finite-difference linear advection solver using periodic boundary conditions.

  • fdadvect.py: a 1-d first-order explicit finite-difference linear advection solver using upwinded differencing.

  • fv_mol.py: a 1-d method-of-lines second-order accurate advection solver.

  • Fortran/:

    • advect.f90: a Fortran implementation of second-order linear advection. This version does piecewise constant, piecewise linear, and piecewise parabolic (PPM) reconstruction.

• basic-numerics

  • orbit-converge.py (and orbit.py): a demonstration of the convergence of different ODE integration methods for the problem of Earth orbiting around the Sun.

• burgers/

  • burgers.py: a 1-d second-order solver for the inviscid Burgers’ equation, with initial conditions corresponding to a shock and a rarefaction.

• compressible/

  • euler.ipynb: a SymPy IPython notebook that derives the eigenvalues and eigenvectors for the Euler equations.

  • riemann-phase.py: a simple script that plots the Hugoniot curves for a compressible Riemann problem (assuming a gamma-law gas)

• diffusion/

  • diffusion-explicit.py: solve the constant-diffusivity diffusion equation explicitly. The method is first-order accurate in time, but second- order in space. A Gaussian profile is diffused--the analytic solution is also a Gaussian.

  • diffusion-implicit.py: solve the constant-diffusivity diffusion equation implicitly. Crank-Nicolson time-discretization is used, resulting in a second-order method. A Gaussian profile is diffused.

• elliptic/

  • poisson_fft.py: an FFT solver for a 2-d Poisson problem with periodic boundaries.

• finite-volume/

  • conservative-interpolation.ipynb: an IPython notebook that illustrates how to derive high-order conservative interpolants for finite-volume data.

• incompressible/

  • project.py: a simple example of using a projection to recover a divergence-free velocity field.

• multigrid/

  • mg_converge.py: a convergence test of the multigrid solver. A Poisson problem is solved at various resolutions and compared to the exact solution. This demonstrates second-order accuracy.

  • mg_test.py: a simple driver for the multigrid solver. This sets up and solves a Poisson problem and plots the behavior of the solution as a function of V-cycle number.

  • multigrid.py: a multigrid class for cell-centered data. This implements pure V-cycles. A square domain with 2 N zones (N a positive integer) is required.

  • patch1d.py: a class for 1-d cell-centered data that lives on a grid. This manages the data, handles boundary conditions, and provides routines for prolongation and restriction to other grids.

• multiphysics/

  • burgersvisc.py: solve the viscous Burgers equation. The advective terms are treated explicitly with a second-order accurate method. The diffusive term is solved using an implicit Crank-Nicolson discretiza- tion. The overall coupling is second-order accurate.

  • diffusion-reaction.py: solve a diffusion-reaction equation that propagates a diffusive reacting front (flame). A simple reaction term is modeled. The diffusion is solved using a second-order Crank-Nicolson discretization. The reactions are evolved using the VODE ODE solver (via SciPy). The two processes are coupled together using Strang-splitting to be second-order accurate in time.

• parallel/

  • relax-mpi.f90: a simple example of pure relaxiation using domain decomposition + message passing (through MPI) to implement smoothing in a parallel fashion.

  • relax-omp.f90: a simple example of pure relaxation using OpenMP to parallelize the loops using shared-memory.