Sponge

Castro uses a sponge source term to damp velocities in regions where we are not interested in the dynamics, to prevent them from dominating the timestep constraint. Often these are buffer regions between the domain of interest and the boundary conditions.

The sponge parameters are set in the inputs file. The timescale of the damping is set through castro.sponge_timescale, while factors such as the radius/density/pressure at which the sponge starts to begin being applied are described below.

The sponge value, \(f_\mathrm{sponge}\) is computed as described below and then the sponge factor, \(f\) is computed as:

\[f = - \left [ 1 - \frac{1}{1 + \alpha f_\mathrm{sponge}}\right ]\]

for an implicit update or

\[f = -\alpha f_\mathrm{sponge}\]

for an explicit update. This choice is controlled by sponge_implicit. Here, \(\alpha\) is constructed from the sponge_timescale parameter, \(t_\mathrm{sponge}\) as:

\[\alpha = \frac{\Delta t}{t_\mathrm{sponge}}\]

The sponge source is then added to the momentum and total energy equations.

The general sponge parameters are:

variable	runtime parameter
\(f_\mathrm{lower}\)	sponge_lower_factor
\(f_\mathrm{upper}\)	sponge_upper_factor

There are three sponges, each controlled by different runtime parameters:

• radial sponge : The radial sponge operates beyond some radius (define in terms of the center(:) position for the problem). It takes the form:

  \[\begin{split}f_\mathrm{sponge} = \left \{ \begin{array}{cc} f_\mathrm{lower} & r < r_\mathrm{lower} \\ f_\mathrm{lower} + \frac{f_\mathrm{upper} - f_\mathrm{lower}}{2} \left [ 1 - \cos \left ( \frac{\pi (r - r_\mathrm{lower})}{\Delta r} \right ) \right ] & r_\mathrm{lower} \le r < r_\mathrm{upper} \\ f_\mathrm{upper} & r \ge r_\mathrm{upper} \end{array} \right .\end{split}\]

  The parameters controlling the various quantities here are:

  variable	runtime parameter
  \(r_\mathrm{lower}\)	sponge_lower_radius
  \(r_\mathrm{upper}\)	sponge_upper_radius

  and \(\Delta r = r_\mathrm{upper} - r_\mathrm{lower}\) .

• density sponge : The density sponge turns on based on density thresholds. At high densities, we usually care about the dynamics, so we will want the sponge off, but as the density lowers, we gradually turn on the sponge. It takes the form:

  \[\begin{split}f_\mathrm{sponge} = \left \{ \begin{array}{cc} f_\mathrm{lower} & \rho > \rho_\mathrm{upper} \\ f_\mathrm{lower} + \frac{f_\mathrm{upper} - f_\mathrm{lower}}{2} \left [ 1 - \cos \left ( \frac{\pi (\rho - \rho_\mathrm{upper})}{\Delta \rho} \right ) \right ] & \rho_\mathrm{upper} \ge \rho > \rho_\mathrm{lower} \\ f_\mathrm{upper} & \rho < \rho_\mathrm{lower} \end{array} \right .\end{split}\]

  The parameters controlling the various quantities here are:

  variable	runtime parameter
  \(\rho_\mathrm{lower}\)	sponge_lower_density
  \(\rho_\mathrm{upper}\)	sponge_upper_density

  and \(\Delta \rho = \rho_\mathrm{upper} - \rho_\mathrm{lower}\) .

• pressure sponge : The pressure sponge is just like the density sponge, except it is keyed on pressure. It takes the form:

  \[\begin{split}f_\mathrm{sponge} = \left \{ \begin{array}{cc} f_\mathrm{lower} & p > p_\mathrm{upper} \\ f_\mathrm{lower} + \frac{f_\mathrm{upper} - f_\mathrm{lower}}{2} \left [ 1 - \cos \left ( \frac{\pi (p - p_\mathrm{upper})}{\Delta p} \right ) \right ] & p_\mathrm{upper} \ge p \ge p_\mathrm{lower} \\ f_\mathrm{upper} & \rho < \rho_\mathrm{lower} \end{array} \right .\end{split}\]

  The parameters controlling the various quantities here are:

  variable	runtime parameter
  \(p_\mathrm{lower}\)	sponge_lower_pressure
  \(p_\mathrm{upper}\)	sponge_upper_pressure

  and \(\Delta p = p_\mathrm{upper} - p_\mathrm{lower}\) .