# compressible package¶

The pyro compressible hydrodynamics solver. This implements the second-order (piecewise-linear), unsplit method of Colella 1990.

## compressible.BC module¶

compressible-specific boundary conditions. Here, in particular, we implement an HSE BC in the vertical direction.

Note: the pyro BC routines operate on a single variable at a time, so some work will necessarily be repeated.

Also note: we may come in here with the aux_data (source terms), so we’ll do a special case for them

compressible.BC.inflow_post_bc(var, g)[source]
compressible.BC.inflow_pre_bc(var, g)[source]
compressible.BC.user(bc_name, bc_edge, variable, ccdata)[source]

A hydrostatic boundary. This integrates the equation of HSE into the ghost cells to get the pressure and density under the assumption that the specific internal energy is constant.

Upon exit, the ghost cells for the input variable will be set

Parameters
bc_name{‘hse’}

The descriptive name for the boundary condition – this allows for pyro to have multiple types of user-supplied boundary conditions. For this module, it needs to be ‘hse’.

bc_edge{‘ylb’, ‘yrb’}

The boundary to update: ylb = lower y boundary; yrb = upper y boundary.

variable{‘density’, ‘x-momentum’, ‘y-momentum’, ‘energy’}

The variable whose ghost cells we are filling

ccdataCellCenterData2d object

The data object

## compressible.derives module¶

compressible.derives.derive_primitives(myd, varnames)[source]

derive desired primitive variables from conserved state

## compressible.eos module¶

This is a gamma-law equation of state: p = rho e (gamma - 1), where gamma is the constant ratio of specific heats.

compressible.eos.dens(gamma, pres, eint)[source]

Given the pressure and the specific internal energy, return the density

Parameters
gammafloat

The ratio of specific heats

presfloat

The pressure

eintfloat

The specific internal energy

Returns
outfloat

The density

compressible.eos.pres(gamma, dens, eint)[source]

Given the density and the specific internal energy, return the pressure

Parameters
gammafloat

The ratio of specific heats

densfloat

The density

eintfloat

The specific internal energy

Returns
outfloat

The pressure

compressible.eos.rhoe(gamma, pres)[source]

Given the pressure, return (rho * e)

Parameters
gammafloat

The ratio of specific heats

presfloat

The pressure

Returns
outfloat

The internal energy density, rho e

## compressible.interface module¶

compressible.interface.artificial_viscosity(ng, dx, dy, cvisc, u, v)[source]

Compute the artifical viscosity. Here, we compute edge-centered approximations to the divergence of the velocity. This follows directly Colella Woodward (1984) Eq. 4.5

data locations:

j+3/2--+---------+---------+---------+
|         |         |         |
j+1  +         |         |         |
|         |         |         |
j+1/2--+---------+---------+---------+
|         |         |         |
j +         X         |         |
|         |         |         |
j-1/2--+---------+----Y----+---------+
|         |         |         |
j-1 +         |         |         |
|         |         |         |
j-3/2--+---------+---------+---------+
|    |    |    |    |    |    |
i-1        i        i+1
i-3/2     i-1/2     i+1/2     i+3/2


X is the location of avisco_x[i,j] Y is the location of avisco_y[i,j]

Parameters
ngint

The number of ghost cells

dx, dyfloat

Cell spacings

cviscfloat

viscosity parameter

u, vndarray

x- and y-velocities

Returns
outndarray, ndarray

Artificial viscosity in the x- and y-directions

compressible.interface.consFlux(idir, gamma, idens, ixmom, iymom, iener, irhoX, nspec, U_state)[source]

Calculate the conservative flux.

Parameters
idirint

Are we predicting to the edges in the x-direction (1) or y-direction (2)?

gammafloat

idens, ixmom, iymom, iener, irhoXint

The indices of the density, x-momentum, y-momentum, internal energy density and species partial densities in the conserved state vector.

nspecint

The number of species

U_statendarray

Conserved state vector.

Returns
outndarray

Conserved flux

compressible.interface.riemann_cgf(idir, ng, idens, ixmom, iymom, iener, irhoX, nspec, lower_solid, upper_solid, gamma, U_l, U_r)[source]

Solve riemann shock tube problem for a general equation of state using the method of Colella, Glaz, and Ferguson. See Almgren et al. 2010 (the CASTRO paper) for details.

The Riemann problem for the Euler’s equation produces 4 regions, separated by the three characteristics (u - cs, u, u + cs):

u - cs    t    u      u + cs
\       ^   .       /
\  *L  |   . *R   /
\     |  .     /
\    |  .    /
L    \   | .   /    R
\  | .  /
\ |. /
\|./
----------+----------------> x


We care about the solution on the axis. The basic idea is to use estimates of the wave speeds to figure out which region we are in, and: use jump conditions to evaluate the state there.

Only density jumps across the u characteristic. All primitive variables jump across the other two. Special attention is needed if a rarefaction spans the axis.

Parameters
idirint

Are we predicting to the edges in the x-direction (1) or y-direction (2)?

ngint

The number of ghost cells

nspecint

The number of species

idens, ixmom, iymom, iener, irhoXint

The indices of the density, x-momentum, y-momentum, internal energy density and species partial densities in the conserved state vector.

lower_solid, upper_solidint

Are we at lower or upper solid boundaries?

gammafloat

U_l, U_rndarray

Conserved state on the left and right cell edges.

Returns
outndarray

Conserved flux

compressible.interface.riemann_hllc(idir, ng, idens, ixmom, iymom, iener, irhoX, nspec, lower_solid, upper_solid, gamma, U_l, U_r)[source]

This is the HLLC Riemann solver. The implementation follows directly out of Toro’s book. Note: this does not handle the transonic rarefaction.

Parameters
idirint

Are we predicting to the edges in the x-direction (1) or y-direction (2)?

ngint

The number of ghost cells

nspecint

The number of species

idens, ixmom, iymom, iener, irhoXint

The indices of the density, x-momentum, y-momentum, internal energy density and species partial densities in the conserved state vector.

lower_solid, upper_solidint

Are we at lower or upper solid boundaries?

gammafloat

U_l, U_rndarray

Conserved state on the left and right cell edges.

Returns
outndarray

Conserved flux

compressible.interface.riemann_prim(idir, ng, irho, iu, iv, ip, iX, nspec, lower_solid, upper_solid, gamma, q_l, q_r)[source]

this is like riemann_cgf, except that it works on a primitive variable input state and returns the primitive variable interface state

Solve riemann shock tube problem for a general equation of state using the method of Colella, Glaz, and Ferguson. See Almgren et al. 2010 (the CASTRO paper) for details.

The Riemann problem for the Euler’s equation produces 4 regions, separated by the three characteristics $$(u - cs, u, u + cs)$$:

u - cs    t    u      u + cs
\       ^   .       /
\  *L  |   . *R   /
\     |  .     /
\    |  .    /
L    \   | .   /    R
\  | .  /
\ |. /
\|./
----------+----------------> x


We care about the solution on the axis. The basic idea is to use estimates of the wave speeds to figure out which region we are in, and: use jump conditions to evaluate the state there.

Only density jumps across the $$u$$ characteristic. All primitive variables jump across the other two. Special attention is needed if a rarefaction spans the axis.

Parameters
idirint

Are we predicting to the edges in the x-direction (1) or y-direction (2)?

ngint

The number of ghost cells

nspecint

The number of species

irho, iu, iv, ip, iXint

The indices of the density, x-velocity, y-velocity, pressure and species fractions in the state vector.

lower_solid, upper_solidint

Are we at lower or upper solid boundaries?

gammafloat

q_l, q_rndarray

Primitive state on the left and right cell edges.

Returns
outndarray

Primitive flux

compressible.interface.states(idir, ng, dx, dt, irho, iu, iv, ip, ix, nspec, gamma, qv, dqv)[source]

predict the cell-centered state to the edges in one-dimension using the reconstructed, limited slopes.

We follow the convection here that V_l[i] is the left state at the i-1/2 interface and V_l[i+1] is the left state at the i+1/2 interface.

We need the left and right eigenvectors and the eigenvalues for the system projected along the x-direction.

Taking our state vector as $$Q = (\rho, u, v, p)^T$$, the eigenvalues are $$u - c$$, $$u$$, $$u + c$$.

We look at the equations of hydrodynamics in a split fashion – i.e., we only consider one dimension at a time.

Considering advection in the x-direction, the Jacobian matrix for the primitive variable formulation of the Euler equations projected in the x-direction is:

    / u   r   0   0 \
| 0   u   0  1/r |
A = | 0   0   u   0  |
\ 0  rc^2 0   u  /


The right eigenvectors are:

     /  1  \        / 1 \        / 0 \        /  1  \
|-c/r |        | 0 |        | 0 |        | c/r |
r1 = |  0  |   r2 = | 0 |   r3 = | 1 |   r4 = |  0  |
\ c^2 /        \ 0 /        \ 0 /        \ c^2 /


In particular, we see from r3 that the transverse velocity (v in this case) is simply advected at a speed u in the x-direction.

The left eigenvectors are:

l1 =     ( 0,  -r/(2c),  0, 1/(2c^2) )
l2 =     ( 1,     0,     0,  -1/c^2  )
l3 =     ( 0,     0,     1,     0    )
l4 =     ( 0,   r/(2c),  0, 1/(2c^2) )


The fluxes are going to be defined on the left edge of the computational zones:

 |             |             |             |
|             |             |             |
-+------+------+------+------+------+------+--
|     i-1     |      i      |     i+1     |
^ ^           ^
q_l,i q_r,i  q_l,i+1


q_r,i and q_l,i+1 are computed using the information in zone i,j.

Parameters
idirint

Are we predicting to the edges in the x-direction (1) or y-direction (2)?

ngint

The number of ghost cells

dxfloat

The cell spacing

dtfloat

The timestep

irho, iu, iv, ip, ixint

Indices of the density, x-velocity, y-velocity, pressure and species in the state vector

nspecint

The number of species

gammafloat

qvndarray

The primitive state vector

dqvndarray

Spatial derivitive of the state vector

Returns
outndarray, ndarray

State vector predicted to the left and right edges

## compressible.simulation module¶

class compressible.simulation.Simulation(solver_name, problem_name, rp, timers=None, data_class=<class 'mesh.patch.CellCenterData2d'>)[source]

The main simulation class for the corner transport upwind compressible hydrodynamics solver

dovis()[source]

Do runtime visualization.

evolve()[source]

Evolve the equations of compressible hydrodynamics through a timestep dt.

initialize(extra_vars=None, ng=4)[source]

Initialize the grid and variables for compressible flow and set the initial conditions for the chosen problem.

method_compute_timestep()[source]

The timestep function computes the advective timestep (CFL) constraint. The CFL constraint says that information cannot propagate further than one zone per timestep.

We use the driver.cfl parameter to control what fraction of the CFL step we actually take.

write_extras(f)[source]

Output simulation-specific data to the h5py file f

class compressible.simulation.Variables(myd)[source]

Bases: object

a container class for easy access to the different compressible variable by an integer key

compressible.simulation.cons_to_prim(U, gamma, ivars, myg)[source]

convert an input vector of conserved variables to primitive variables

compressible.simulation.prim_to_cons(q, gamma, ivars, myg)[source]

convert an input vector of primitive variables to conserved variables

## compressible.unsplit_fluxes module¶

Implementation of the Colella 2nd order unsplit Godunov scheme. This is a 2-dimensional implementation only. We assume that the grid is uniform, but it is relatively straightforward to relax this assumption.

There are several different options for this solver (they are all discussed in the Colella paper).

• limiter: 0 = no limiting; 1 = 2nd order MC limiter; 2 = 4th order MC limiter

• riemann: HLLC or CGF (for Colella, Glaz, and Freguson solver)

• use_flattening: set to 1 to use the multidimensional flattening at shocks

• delta, z0, z1: flattening parameters (we use Colella 1990 defaults)

The grid indices look like:

j+3/2--+---------+---------+---------+
|         |         |         |
j+1 _|         |         |         |
|         |         |         |
|         |         |         |
j+1/2--+---------XXXXXXXXXXX---------+
|         X         X         |
j _|         X         X         |
|         X         X         |
|         X         X         |
j-1/2--+---------XXXXXXXXXXX---------+
|         |         |         |
j-1 _|         |         |         |
|         |         |         |
|         |         |         |
j-3/2--+---------+---------+---------+
|    |    |    |    |    |    |
i-1        i        i+1
i-3/2     i-1/2     i+1/2     i+3/2


We wish to solve

$U_t + F^x_x + F^y_y = H$

we want U_{i+1/2}^{n+1/2} – the interface values that are input to the Riemann problem through the faces for each zone.

Taylor expanding yields:

 n+1/2                     dU           dU
U          = U   + 0.5 dx  --  + 0.5 dt --
i+1/2,j,L    i,j          dx           dt

dU             dF^x   dF^y
= U   + 0.5 dx  --  - 0.5 dt ( ---- + ---- - H )
i,j          dx              dx     dy

dU      dF^x            dF^y
= U   + 0.5 ( dx -- - dt ---- ) - 0.5 dt ---- + 0.5 dt H
i,j           dx       dx              dy

dt       dU           dF^y
= U   + 0.5 dx ( 1 - -- A^x ) --  - 0.5 dt ---- + 0.5 dt H
i,j               dx       dx            dy

dt       _            dF^y
= U   + 0.5  ( 1 - -- A^x ) DU  - 0.5 dt ---- + 0.5 dt H
i,j             dx                     dy

+----------+-----------+  +----+----+   +---+---+
|                   |            |

this is the monotonized   this is the   source term
central difference term   transverse
flux term


There are two components, the central difference in the normal to the interface, and the transverse flux difference. This is done for the left and right sides of all 4 interfaces in a zone, which are then used as input to the Riemann problem, yielding the 1/2 time interface values:

 n+1/2
U
i+1/2,j


Then, the zone average values are updated in the usual finite-volume way:

 n+1    n     dt    x  n+1/2       x  n+1/2
U    = U    + -- { F (U       ) - F (U       ) }
i,j    i,j   dx       i-1/2,j        i+1/2,j

dt    y  n+1/2       y  n+1/2
+ -- { F (U       ) - F (U       ) }
dy       i,j-1/2        i,j+1/2


Updating U_{i,j}:

• We want to find the state to the left and right (or top and bottom) of each interface, ex. U_{i+1/2,j,[lr]}^{n+1/2}, and use them to solve a Riemann problem across each of the four interfaces.

• U_{i+1/2,j,[lr]}^{n+1/2} is comprised of two parts, the computation of the monotonized central differences in the normal direction (eqs. 2.8, 2.10) and the computation of the transverse derivatives, which requires the solution of a Riemann problem in the transverse direction (eqs. 2.9, 2.14).

• the monotonized central difference part is computed using the primitive variables.

• We compute the central difference part in both directions before doing the transverse flux differencing, since for the high-order transverse flux implementation, we use these as the input to the transverse Riemann problem.

compressible.unsplit_fluxes.unsplit_fluxes(my_data, my_aux, rp, ivars, solid, tc, dt)[source]

unsplitFluxes returns the fluxes through the x and y interfaces by doing an unsplit reconstruction of the interface values and then solving the Riemann problem through all the interfaces at once

currently we assume a gamma-law EOS

The runtime parameter grav is assumed to be the gravitational acceleration in the y-direction

Parameters
my_dataCellCenterData2d object

The data object containing the grid and advective scalar that we are advecting.

rpRuntimeParameters object

The runtime parameters for the simulation

varsVariables object

The Variables object that tells us which indices refer to which variables

tcTimerCollection object

The timers we are using to profile

dtfloat

The timestep we are advancing through.

Returns
outndarray, ndarray

The fluxes on the x- and y-interfaces