A truly Newtonian softening length for disc simulations

TL;DR

This paper derives a physically consistent method for choosing the softening length in disc simulations, improving accuracy by accounting for cell shape and proposing a universal prescription.

Contribution

It introduces a novel, shape-dependent softening length formula for disc simulations that enhances accuracy over traditional fixed fractions of disc thickness.

Findings

The softening length depends critically on cell shape and can be real or imaginary.

Using a fixed fraction of disc thickness is suboptimal.

The new prescription improves numerical precision by 2-3 digits.

Abstract

The softened point mass model is commonly used in simulations of gaseous discs including self-gravity while the value of associated length \lambda remains, to some degree, controversial. This ``parameter'' is however fully constrained when, in a discretized disc, all fluid cells are demanded to obey Newton's law. We examine the topology of solutions in this context, focusing on cylindrical cells more or less vertically elongated. We find that not only the nominal length depends critically on the cell's shape (curvature, radial extension, height), but it is either a real or an imaginary number. Setting \lambda as a fraction of the local disc thickness -- as usually done -- is indeed not the optimal choice. We then propose a novel prescription valid irrespective of the disc properties and grid spacings. The benefit, which amounts to 2-3 more digits typically, is illustrated in a few concrete cases. A detailed mathematical analysis is in progress.