Small dust grain dynamics on adaptive mesh-refinement grids. I. Methods
  Small dust grains are essential ingredients of star, disk and planet
formation. We present an Eulerian numerical approach to study small dust grains
dynamics in the context of star and protoplanetary disk formation. It is
designed for finite volume codes. We use it to investigate dust dynamics during
the protostellar collapse. We present a method to solve the monofluid equations
of gas and dust mixtures with several dust species in the diffusion
approximation implemented in the adaptive-mesh-refinement code RAMSES. It uses
a finite volume second-order Godunov method with a predictor-corrector MUSCL
scheme to estimate the fluxes between the grid cells. We benchmark our method
against six distinct tests, dustyadvect, dustydiffuse, dustyshock, dustywave,
settling and dustycollapse. We show that the scheme is second-order accurate in
space on uniform grids and intermediate between second and first-order on
non-uniform grids. We apply our method on various dustycollapse simulations of
1 solar mass cores composed of gas and dust. We developed an efficient approach
to treat gas and dust dynamics in the diffusion regime on grid based codes. The
canonical tests have been successfully passed. In the context of protostellar
collapse, we show that dust is less coupled to the gas in the outer regions of
the collapse where grains larger than approximately 100 micrometers fall
significantly faster than the gas.
