  Approaches used in modern numerical simulations of the dynamics of dust and
gas in circumstellar disks are tested. The gas and dust are treated like
interpenetrating continuous media that can exchange momentum. A stiff coupling
between the gas and dust phases is typical for such disks, with the dust
stopping time much less than the characteristic dynamical time scale. This
imposes high demands on the methods used to simulate the dust dynamics. A grid,
piecewise-parabolic method is used as the basic algorithm for solving the
gas-dynamical equations. Numerical solutions obtained using various methods to
compute the momentum exchanges are presented for the case of monodisperse dust.
Numerical solutions are obtained for shock tube problem and the propagation of
sound waves in a gas-dust medium. The studied methods are compared in terms of
their ability to model media with (a) an arbitrary (short or long) dust
stopping time, and (b) an arbitrary dust concentration in the gas (varying the
dust to gas mass ratio from 0.01 to 1). A method for computing the momentum
exchange with infinite-order accuracy in time is identified, which makes it
possible to satisfy the conditions (a) and (b) with minimal computational
costs. A first-order method that shows similar results in the test computations
is also presented. It is shown that the proposed first-order method for
monodisperse dust can be extended to a regime when the dust is polydisperse;
i.e., a regime represented by several fractions with different stopping times.
Formulas for computing the gas and dust velocities for polydisperse dust with
each fraction exchanging momentum with the gas are presented.
