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Compressible hydrodynamics

The Euler equations of compressible hydrodynamics express conservation of mass, momentum, and energy. For the conserved state, $Uc = (rho, rho Ub, rho E)^intercal$, the conserved system can be written as:

\Uc_t + \nabla \cdot {\bf F}(\Uc) = {\bf S}

where ${bf F}$ are the fluxes and ${bf S}$ are any source terms. In terms of components, they take the form (with a gravity source term):

\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \Ub) &= 0 \\
\frac{\partial (\rho \Ub)}{\partial t} + \nabla \cdot (\rho \Ub \Ub) + \nabla p &= \rho {\bf g} \\
\frac{\partial (\rho E)}{\partial t} + \nabla \cdot [(\rho E + p ) \Ub] &= \rho \Ub \cdot {\bf g}

with \rho E = \rho e + \frac{1}{2} \rho |\Ub|^2. The system is closed with an equation of state of the form:

p = p(\rho, e)

Note

The Euler equations do not include any dissipation terms, since they are usually negligible in astrophysics.

pyro has several compressible solvers to solve this equation set. The implementations here have flattening at shocks, artificial viscosity, a simple gamma-law equation of state, and (in some cases) a choice of Riemann solvers. Optional constant gravity in the vertical direction is allowed.

Note

All the compressible solvers share the same problems/ directory, which lives in compressible/problems/. For the other compressible solvers, we simply use a symbolic-link to this directory in the solver's directory.

compressible solver

:py:mod:`pyro.compressible` is based on a directionally unsplit (the corner transport upwind algorithm) piecewise linear method for the Euler equations, following :cite:`colella:1990`. This is overall second-order accurate.

The parameters for this solver are:

compressible_rk solver

:py:mod:`pyro.compressible_rk` uses a method of lines time-integration approach with piecewise linear spatial reconstruction for the Euler equations. This is overall second-order accurate.

The parameters for this solver are:

compressible_fv4 solver

:py:mod:`pyro.compressible_fv4` uses a 4th order accurate method with RK4 time integration, following :cite:`mccorquodalecolella`.

The parameter for this solver are:

compressible_sdc solver

:py:mod:`pyro.compressible_sdc` uses a 4th order accurate method with spectral-deferred correction (SDC) for the time integration. This shares much in common with the :py:mod:`pyro.compressible_fv4` solver, aside from how the time-integration is handled.

The parameters for this solver are:

.. toctree::
   :maxdepth: 1
   :hidden:

   compressible_problems
   compressible_sources
   compressible_exercises