  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, whose magnitude is often unknown or poorly
constrained. With the current trend of simulating accretion disks with no or
very low prescribed physical viscosity, it becomes essential to understand and
quantify this inherent numerical diffusion, in the form of a numerical
viscosity. We study the behavior of the viscous spreading ring and the spiral
instability that develops in it. We then use this setup to quantify the
numerical viscosity in Cartesian grids and study its properties. We simulate
the viscous spreading ring and the related instability on a two-dimensional
polar grid using PLUTO as well as FARGO, and ensure convergence of our results
with a resolution study. We then repeat our models on a Cartesian grid and
measure the numerical viscosity by comparing results to the known analytical
solution, using PLUTO and Athena++. We find that the numerical viscosity in a
Cartesian grid scales with resolution as approximately $\nu_{num}\propto\Delta
x^2$ and is equivalent to an effective $\alpha\sim10^{-4}$ for a common
numerical setup. We also show 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. 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.
