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 discretization 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 percent 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 minimizes the numerical cost while 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.