Evolving grain-size distributions embedded in gas flows
  We present a numerical approach for accurately evolving a dust grain-size
distribution undergoing number-conserving (such as sputtering) and/or
mass-conserving (such as shattering) processes. As typically observed
interstellar dust distributions follow a power-law, our method adopts a
power-law discretisation and uses both the grain mass and number densities in
each bin to determine the power-law parameters. This power-law method is
complementary to piecewise-constant and linear methods in the literature. We
find that the power-law method surpasses the other two approaches, especially
for small bin numbers. In the sputtering tests the relative error in the total
grain mass remains below 0.01% independent of the number of bins N, while the
other methods only achieve this for N > 50 or higher. Likewise, the shattering
test shows that the method also produces small relative errors in the total
grain numbers while conserving mass. Not only does the power-law method
conserve the global distribution properties, it also preserves the inter-bin
characteristics so that the shape of the distribution is recovered to a high
degree. This does not always happen for the constant and linear methods,
especially not for small bin numbers. Implementing the power-law method in a
hydrodynamical code thus minimises the numerical cost whilst maintaining high
accuracy. The method is not limited to dust grain distributions, but can also
be applied to the evolution of any distribution function, such as a cosmic-ray
distribution affected by synchrotron radiation or inverse-Compton scattering.
