Measuring the numerical viscosity in simulations of protoplanetary disks in Cartesian grids

The viscously spreading ring revisited

¹ Institut für Astronomie und Astrophysik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
² Astronomy Unit, Department of Physics and Astronomy, Queen Mary University of London, London E1 4NS, UK
e-mail: a.ziampras@qmul.ac.uk; r.p.nelson@qmul.ac.uk
³ Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany
e-mail: jjoseph@aip.de

Received: 1 December 2022
Accepted: 24 July 2023

Abstract

Context. Hydrodynamical simulations solve the governing equations on a discrete grid of space and time. This discretization causes numerical diffusion similar to a physical viscous diffusion, the magnitude of which is often unknown or poorly constrained. With the current trend of simulating accretion disks with no or very low prescribed physical viscosity, it has become essential to understand and quantify this inherent numerical diffusion in the form of a numerical viscosity.

Aims. We study the behavior of the viscous spreading ring and the spiral instability that develops in it. We aim to use this setup to quantify the numerical viscosity in Cartesian grids and study its properties.

Methods. We simulated the viscous spreading ring and the related instability on a two-dimensional polar grid using PLUTO as well as FARGO, ensuring the convergence of our results with a resolution study. We then repeated our models on a Cartesian grid and measured the numerical viscosity by comparing results to the known analytical solution using PLUTO and Athena++.

Results. We find that the numerical viscosity in a Cartesian grid scales with resolution as approximately v_num ∝ Δx² and is equivalent to an effective α ~ 10⁻⁴ for a common numerical setup. We also showed that the spiral instability manifests as a single leading spiral throughout the whole domain on polar grids. This is contrary to previous results and indicates that sufficient resolution is necessary in order to correctly resolve the instability.

Conclusions. Our results are relevant in the context of models where the origin should be included in the computational domain, or when polar grids cannot be used. Examples of such cases include models of disk accretion onto a central binary and, inherently, Cartesian codes.