Carbuncle Instability

Radiative shocks in particular are susceptible to the carbuncle instability. There are several ways to combat this instability by adding the appropriate amount of diffusion. Sanders et al. 1998 (JCompPhys 145:511) describe a method to increase diffusion near carbuncles by modifying the wave speeds used in the Roe solver in one of three ways:

begin{displaymath} |\Lambda_l| = |\Lambda_l| + \eta \mbox{ (a)} \end{displaymath} \begin{displaymath} |\Lambda_l| = \max(|\Lambda_l|, \eta) \mbox{ (b)} \end{displaymath} \begin{displaymath} |\Lambda_l| = \left\{ \begin{array}{lr} |\Lambda_l| & |\Lambda_l| > 2\eta\\ \frac{|\Lambda_l|^2}{4\eta} + \eta & \mbox{otherwise} \end{array} \right. \mbox{ (c)} \end{displaymath}

where \eta = \frac{1}{2}\displaystyle\max_l \left(|\lambda_l(U_R)-\lambda_l(U_L)|\right)

Note in Athena's method paper they suggest using \eta = \frac{1}{2}|\displaystyle\max_l(\lambda_l(U_R))-\displaystyle\min_l(\lambda_l(U_L))| along with option b, however for subsonic smooth flows this would add additional dissipation. Additionally option b from Sanders et al. does not seems to be invariant under Galilean transformations.

In AstroBEAR when ViscCD == 1, option (a) is used combined with Sander's definition of \eta.

In AstroBEAR when ViscCD == 2, H-viscosity is turned on. For H-viscosity, \eta is replaced with the maximum value of \eta from the surrounding interfaces in the shape of on H.

While the original formalism in Sander's paper was for modifying the wave speeds used by the Roe Solver - HLL type solvers can also be modified by adjusting the slowest and the fastest wave in a similar fashion.

Note that currently the exact Riemann solver does not support ViscCD > 0

Alternatively, Sutherland et al. 2003 (ApJs 147:187) suggested using a local oscillation filter to add dissipative terms only in regions unstable to carbuncles. A mask that checks for density striations normal to slope limited regions (presumably near shocks) triggers additional transverse fluxes that damp the instability. As a result, cells that are over/under dense and masked as having carbuncles are partially averaged with its under/over dense neighbors where the degree of averaging is parametrized by \alpha_{LOF} which is set to .075 by default, but can be adjusted in solver.data

The image below shows a snapshot from a simulation where the top section is of a run with ViscCD == 2, the middle section is a run with ViscCD == 0, and the bottom section is a run using only the LOF.

For movies comparing these different viscosities see ​http://www.pas.rochester.edu/~johannjc/Meeting/main.html?dir=110329