A Staggered Semi-Analytic Method for Simulating Dust Grains Subject to Gas Drag

TL;DR

The paper introduces the staggered semi-analytic (SSA) method for simulating dust-gas interactions in astrophysics, offering higher accuracy and stability for large time steps compared to existing methods.

Contribution

The SSA method is a new, easy-to-implement approach that improves accuracy and stability in stiff dust-gas simulations, outperforming previous implicit and semi-analytic techniques.

Findings

SSA achieves orders-of-magnitude higher accuracy.

SSA allows for time steps 100 times larger.

SSA is second-order accurate and symplectic in certain regimes.

Abstract

Numerical simulations of dust-gas dynamics are one of the fundamental tools in astrophysical research, such as the study of star and planet formation. It is common to find tightly coupled dust and gas in astrophysical systems, which demands that any practical integration method be able to take time steps $\Delta t$ much longer than the stopping time $t_{\rm s}$ due to drag. A number of methods have been developed to ensure stability in this stiff ($\Delta t\gg t_{\rm s}$) regime, but there remains large room for improvement in terms of accuracy. In this paper, we describe an easy-to-implement method, the "staggered semi-analytic method" (SSA), and conduct numerical tests to compare it to other implicit and semi-analytic methods, including the $2^{\rm nd}$ order implicit method and the Verlet method. SSA makes use of a staggered step to better approximate the terminal velocity in the stiff regime. In applications to protoplanetary disks, this not only leads to orders-of-magnitude higher accuracy than the other methods, but also provides greater stability, making it possible to take time steps 100 times larger in some situations. SSA is also $2^{\rm nd}$ order accurate and symplectic when $\Delta t \ll t_{\rm s}$. More generally, the robustness of SSA makes it applicable to linear dust-gas drag in virtually any context.