A Fast Second-order Solver for Stiff Multifluid Dust and Gas Hydrodynamics

Abstract

Abstract We present MDIRK: a multifluid second-order diagonally implicit Runge–Kutta method to study momentum transfer between gas and an arbitrary number (N) of dust species. The method integrates the equations of hydrodynamics with an implicit–explicit scheme and solves the stiff source term in the momentum equation with a diagonally implicit, asymptotically stable Runge–Kutta method (DIRK). In particular, DIRK admits a simple analytical solution that can be evaluated with O ( N ) operations, instead of standard matrix inversion, which is O ( N ) 3 . Therefore, the analytical solution significantly reduces the computational cost of the multifluid method, making it suitable for studying the dynamics of systems with particle-size distributions. We demonstrate that the method conserves momentum to machine precision and converges to the correct equilibrium solution with constant external acceleration. To validate our numerical method we present a series of simple hydrodynamic tests, including damping of sound waves, dusty shocks, a multifluid dusty Jeans instability, and a steady-state gas–dust drift calculation. The simplicity of MDIRK lays the groundwork to build fast high-order, asymptotically stable multifluid methods.