A Fast second-order solver for stiff multifluid dust and gas hydrodynamics

TL;DR

The paper introduces MDIRK, an efficient second-order implicit Runge-Kutta method for simulating stiff multifluid hydrodynamics involving gas and multiple dust species, reducing computational costs while maintaining accuracy.

Contribution

It develops a novel diagonally-implicit Runge-Kutta scheme that simplifies solving stiff source terms with an analytical solution, enabling faster simulations of multifluid systems.

Findings

Conserves momentum to machine precision.

Accurately models damping of sound waves and shocks.

Validates with multiple hydrodynamic tests.

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 (IMEX) 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 $\mathcal{O}(N)$ operations, instead of standard matrix inversion, which is $\mathcal{O}(N)^3$. 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 multi-fluid 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.