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

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.