Compressible hydrodynamics

The Euler equations of compressible hydrodynamics take the form:

\[\begin{split}\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho U) &= 0 \\ \frac{\partial (\rho U)}{\partial t} + \nabla \cdot (\rho U U) + \nabla p &= \rho g \\ \frac{\partial (\rho E)}{\partial t} + \nabla \cdot [(\rho E + p ) U] &= \rho U \cdot g\end{split}\]

with \(\rho E = \rho e + \frac{1}{2} \rho |U|^2\) and \(p = p(\rho, e)\). Note these 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

pyro.compressible is based on a directionally unsplit (the corner transport upwind algorithm) piecewise linear method for the Euler equations, following [Colella90]. This is overall second-order accurate.

The parameters for this solver are:

• section: [compressible]

  option	value	description
  use_flattening	1	apply flattening at shocks (1)
  z0	0.75	flattening z0 parameter
  z1	0.85	flattening z1 parameter
  delta	0.33	flattening delta parameter
  cvisc	0.1	artificial viscosity coefficient
  limiter	2	limiter (0 = none, 1 = 2nd order, 2 = 4th order)
  grav	0.0	gravitational acceleration (in y-direction)
  riemann	HLLC	HLLC or CGF

• section: [driver]

  option	value	description
  cfl	0.8

• section: [eos]

  option	value	description
  gamma	1.4	pres = rho ener (gamma - 1)

• section: [particles]

  option	value	description
  do_particles	0
  particle_generator	grid

supported problems

acoustic_pulse

The acoustic pulse problem described in McCorquodale & Colella 2011. This uses a uniform background and a small pressure perturbation that drives a low Mach number soundwave. This problem is useful for testing convergence of a compressible solver.

parameters:

name	default
acoustic_pulse.rho0	1.4
acoustic_pulse.drho0	0.14

advect

A simple advection test. A density perturbation is set with a constant pressure in the domain and a velocity field is set to advect the profile across the domain. This is useful for testing convergence.

bubble

A buoyant perturbation (bubble) is placed in an isothermal hydrostatic atmosphere (plane-parallel). It will rise and deform (due to shear) parameters:

name	default
bubble.dens_base	10.0
bubble.scale_height	2.0
bubble.x_pert	2.0
bubble.y_pert	2.0
bubble.r_pert	0.25
bubble.pert_amplitude_factor	5.0
bubble.dens_cutoff	0.01

convection

A heat source in a layer at some height above the bottom will drive convection in an adiabatically stratified atmosphere. parameters:

name	default
convection.dens_base	10.0
convection.scale_height	4.0
convection.y_height	2.0
convection.thickness	0.25
convection.e_rate	0.1
convection.dens_cutoff	0.01

gresho

The Gresho vortex problem sets up a toroidal velocity field that is balanced by a radial pressure gradient. This is in equilibrium and the state should remain unchanged in time. This version of the problem is based on Miczek, Roepke, and Edelmann 2014. parameters:

name	default
gresho.rho0	1.0
gresho.r	0.2
gresho.mach	0.1
gresho.t_r	1.0

heating

A test of the energy sources. This uses a uniform domain and slowly adds heat to the center over time. parameters:

name	default
heating.rho_ambient	1.0
heating.p_ambient	10.0
heating.r_src	0.1
heating.e_rate	0.1

hse

Initialize an isothermal hydrostatic atmosphere. It should remain static. This is a test of our treatment of the gravitational source term. parameters:

name	default
hse.dens0	1.0
hse.h	1.0

kh

A Kelvin-Helmholtz shear problem. There are 2 shear layers, with the and an optional vertical bulk velocity. This can be used to test the numerical dissipation in the solver. This setup is based on McNally et al. 2012. parameters:

name	default
kh.rho_1	1.0
kh.u_1	-1.0
kh.rho_2	2.0
kh.u_2	1.0
kh.bulk_velocity	0.0

logo

Generate the pyro logo! The word “pyro” is written in the center of the domain and perturbations are placed in the 4 corners to drive converging shocks inward to scramble the logo.

plume

A heat source at a point creates a plume that buoynantly rises in an adiabatically stratified atmosphere. parameters:

name	default
plume.dens_base	10.0
plume.scale_height	4.0
plume.x_pert	2.0
plume.y_pert	2.0
plume.r_pert	0.25
plume.e_rate	0.1
plume.dens_cutoff	0.01

quad

The quadrant problem from Shulz-Rinne et al. 1993; Lax and Lui 1998. Four different states are initialized in the quadrants of the domain, driving shocks and other hydrodynamic waves at the interfaces. This can be used to test the symmetry of the solver.

parameters:

name	default
quadrant.rho1	1.5
quadrant.u1	0.0
quadrant.v1	0.0
quadrant.p1	1.5
quadrant.rho2	0.532258064516129
quadrant.u2	1.206045378311055
quadrant.v2	0.0
quadrant.p2	0.3
quadrant.rho3	0.137992831541219
quadrant.u3	1.206045378311055
quadrant.v3	1.206045378311055
quadrant.p3	0.029032258064516
quadrant.rho4	0.532258064516129
quadrant.u4	0.0
quadrant.v4	1.206045378311055
quadrant.p4	0.3
quadrant.cx	0.5
quadrant.cy	0.5

ramp

A shock hitting a ramp at an oblique angle. This is based on Woodward & Colella 1984. parameters:

name	default
ramp.rhol	8.0
ramp.ul	7.1447096
ramp.vl	-4.125
ramp.pl	116.5
ramp.rhor	1.4
ramp.ur	0.0
ramp.vr	0.0
ramp.pr	1.0

rt

A single-mode Rayleigh-Taylor instability. parameters:

name	default
rt.dens1	1.0
rt.dens2	2.0
rt.amp	1.0
rt.sigma	0.1
rt.p0	10.0

rt2

A RT problem with two distinct modes: short wavelength on the left and long wavelength on the right. This allows one to see how the growth rate depends on wavenumber.

parameters:

name	default
rt2.dens1	1.0
rt2.dens2	2.0
rt2.amp	1.0
rt2.sigma	0.1
rt2.p0	10.0

sedov

The classic Sedov problem. parameters:

name	default
sedov.r_init	0.1
sedov.nsub	4

sod

A general shock tube problem for comparing the solver to an exact Riemann solution. parameters:

name	default
sod.direction	x
sod.dens_left	1.0
sod.dens_right	0.125
sod.u_left	0.0
sod.u_right	0.0
sod.p_left	1.0
sod.p_right	0.1

test

A setup intended for unit testing.

compressible_rk solver

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:

• section: [compressible]

  option	value	description
  use_flattening	1	apply flattening at shocks (1)
  z0	0.75	flattening z0 parameter
  z1	0.85	flattening z1 parameter
  delta	0.33	flattening delta parameter
  cvisc	0.1	artificial viscosity coefficient
  limiter	2	limiter (0 = none, 1 = 2nd order, 2 = 4th order)
  temporal_method	RK4	integration method (see mesh/integration.py)
  grav	0.0	gravitational acceleration (in y-direction)
  riemann	HLLC	HLLC or CGF
  well_balanced	0	use a well-balanced scheme to keep the model in hydrostatic equilibrium

• section: [driver]

  option	value	description
  cfl	0.8

• section: [eos]

  option	value	description
  gamma	1.4	pres = rho ener (gamma - 1)

supported problems

compressible_rk uses the problems defined by compressible.

compressible_fv4 solver

pyro.compressible_fv4 uses a 4th order accurate method with RK4 time integration, following [McCorquodaleColella11].

The parameter for this solver are:

• section: [compressible]

  option	value	description
  use_flattening	1	apply flattening at shocks (1)
  z0	0.75	flattening z0 parameter
  z1	0.85	flattening z1 parameter
  delta	0.33	flattening delta parameter
  cvisc	0.1	artificial viscosity coefficient
  limiter	2	limiter (0 = none, 1 = 2nd order, 2 = 4th order)
  temporal_method	RK4	integration method (see mesh/integration.py)
  grav	0.0	gravitational acceleration (in y-direction)
  riemann	CGF

• section: [driver]

  option	value	description
  cfl	0.8

• section: [eos]

  option	value	description
  gamma	1.4	pres = rho ener (gamma - 1)

supported problems

compressible_fv4 uses the problems defined by compressible.

compressible_sdc solver

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 pyro.compressible_fv4 solver, aside from how the time-integration is handled.

The parameters for this solver are:

• section: [compressible]

  option	value	description
  use_flattening	1	apply flattening at shocks (1)
  z0	0.75	flattening z0 parameter
  z1	0.85	flattening z1 parameter
  delta	0.33	flattening delta parameter
  cvisc	0.1	artificial viscosity coefficient
  limiter	2	limiter (0 = none, 1 = 2nd order, 2 = 4th order)
  temporal_method	RK4	integration method (see mesh/integration.py)
  grav	0.0	gravitational acceleration (in y-direction)
  riemann	CGF

• section: [driver]

  option	value	description
  cfl	0.8

• section: [eos]

  option	value	description
  gamma	1.4	pres = rho ener (gamma - 1)

supported problems

compressible_sdc uses the problems defined by compressible.

Example problems

Note

The 4th-order accurate solver (pyro.compressible_fv4) requires that the initialization create cell-averages accurate to 4th-order. To allow for all the solvers to use the same problem setups, we assume that the initialization routines initialize cell-centers (which is fine for 2nd-order accuracy), and the preevolve() method will convert these to cell-averages automatically after initialization.

Sod

The Sod problem is a standard hydrodynamics problem. It is a one-dimensional shock tube (two states separated by an interface), that exhibits all three hydrodynamic waves: a shock, contact, and rarefaction. Furthermore, there are exact solutions for a gamma-law equation of state, so we can check our solution against these exact solutions. See Toro’s book for details on this problem and the exact Riemann solver.

Because it is one-dimensional, we run it in narrow domains in the x- or y-directions. It can be run as:

pyro_sim.py compressible sod inputs.sod.x
pyro_sim.py compressible sod inputs.sod.y

A simple script, sod_compare.py in analysis/ will read a pyro output file and plot the solution over the exact Sod solution. Below we see the result for a Sod run with 128 points in the x-direction, gamma = 1.4, and run until t = 0.2 s.

We see excellent agreement for all quantities. The shock wave is very steep, as expected. The contact wave is smeared out over ~5 zones—this is discussed in the notes above, and can be improved in the PPM method with contact steepening.

Sedov

The Sedov blast wave problem is another standard test with an analytic solution (Sedov 1959). A lot of energy is point into a point in a uniform medium and a blast wave propagates outward. The Sedov problem is run as:

pyro_sim.py compressible sedov inputs.sedov

The video below shows the output from a 128 x 128 grid with the energy put in a radius of 0.0125 surrounding the center of the domain. A gamma-law EOS with gamma = 1.4 is used, and we run until 0.1

We see some grid effects because it is hard to initialize a small circular explosion on a rectangular grid. To compare to the analytic solution, we need to radially bin the data. Since this is a 2-d explosion, the physical geometry it represents is a cylindrical blast wave, so we compare to Sedov’s cylindrical solution. The radial binning is done with the sedov_compare.py script in analysis/

This shows good agreement with the analytic solution.

quad

The quad problem sets up different states in four regions of the domain and watches the complex interfaces that develop as shocks interact. This problem has appeared in several places (and a detailed investigation is online by Pawel Artymowicz). It is run as:

pyro_sim.py compressible quad inputs.quad

rt

The Rayleigh-Taylor problem puts a dense fluid over a lighter one and perturbs the interface with a sinusoidal velocity. Hydrostatic boundary conditions are used to ensure any initial pressure waves can escape the domain. It is run as:

pyro_sim.py compressible rt inputs.rt

bubble

The bubble problem initializes a hot spot in a stratified domain and watches it buoyantly rise and roll up. This is run as:

pyro_sim.py compressible bubble inputs.bubble

The shock at the top of the domain is because we cut off the stratified atmosphere at some low density and the resulting material above that rains down on our atmosphere. Also note the acoustic signal propagating outward from the bubble (visible in the U and e panels).

Exercises

Explorations

• Measure the growth rate of the Rayleigh-Taylor instability for different wavenumbers.

• There are multiple Riemann solvers in the compressible algorithm. Run the same problem with the different Riemann solvers and look at the differences. Toro’s text is a good book to help understand what is happening.

• Run the problems with and without limiting—do you notice any overshoots?

Extensions

• Limit on the characteristic variables instead of the primitive variables. What changes do you see? (the notes show how to implement this change.)

• Add passively advected species to the solver.

• Add an external heating term to the equations.

• Add 2-d axisymmetric coordinates (r-z) to the solver. This is discussed in the notes. Run the Sedov problem with the explosion on the symmetric axis—now the solution will behave like the spherical sedov explosion instead of the cylindrical explosion.

• Swap the piecewise linear reconstruction for piecewise parabolic (PPM). The notes and the Miller and Colella paper provide a good basis for this. Research the Roe Riemann solver and implement it in pyro.