The impact of numerical oversteepening on the fragmentation boundary in self-gravitating disks

Abstract

Context. Whether or not a self-gravitating accretion disk fragments is still an open issue. There are many different physical and numerical explanations for fragmentation, but simulations often show a non-convergent behavior for ever better resolution. Aims. We aim to investigate the influence of different numerical limiters in Godunov type schemes on the fragmentation boundary in self-gravitating disks. Methods. We have compared the linear and non-linear outcomes in two-dimensional shearingsheet simulations using the VANLEER and the SUPERBEE limiter. Results. We show that choosing inappropriate limiting functions to handle shock-capturing in Godunov type schemes can lead to an overestimation of the surface density in regions with shallow density gradients. The effect amplifies itself on timescales comparable to the dynamical timescale even at high resolutions. This is exactly the environment in which clumps are expected to form. The effect is present without, but scaled up by, self-gravity and also does not depend on cooling. Moreover it can be backtracked to a well known effect called oversteepening. If the effect is also observed in the linear case, the fragmentation limit is shifted to larger values of the critical cooling timescale.

On the diversity and statistical properties of protostellar discs

“…This resolution is sufficient to resolve the local Jeans mass throughout the calculation, which is necessary to model fragmentation of collapsing molecular clouds correctly (Bate & Burkert 1997;Truelove et al 1997;Whitworth 1998;Boss et al 2000;Hubber, Goodwin & Whitworth 2006). More recently, there has been much discussion in the literature about the resolution necessary to resolve fragmentation in isolated gravitationally unstable discs (Nelson 2006;Meru & Bate 2011, 2012Hopkins & Christiansen 2013;Rice et al 2014;Young & Clarke 2015Lin & Kratter 2016;Takahashi, Tsukamoto & Inutsuka 2016;Baehr, Klahr & Kratter 2017;Deng, Mayer & Meru 2017;Klee et al 2017). As yet, there is no consensus as to the resolution that is necessary and sufficient to capture fragmentation of such discs.…”

Closing the gap to convergence of gravitoturbulence in local simulations

“…However, most simulations were again carried out using the Superbee limiter (Roe & Baines 1982) for comparison. Additionally, we use Dormand-Prince time stepping (Dormand & Prince 1980), which showed the best results in the epicyclic motion test (Klee et al 2017).…”

“…We rotate the computational domain by 90 • in contrast to Klee et al (2017). This is advantageous in order to have the first dimension completely accessible for the parallel Fourier solver, while preserving vectorization.…”

“…However, simulations show also fragmentation for larger β, while going to higher resolutions (Meru & Bate 2011;Paardekooper et al 2011). This non-convergence also takes place in the local-approximation (Paardekooper 2012;Baehr & Klahr 2015;Klee et al 2017). Even by using an altered cooling at small scales, Baehr & Klahr (2015) suffer from fragmentation in their high-resolution runs.…”

“…This can be circumvented by using meshless methods. But also grid based codes with shock capturing mechanisms can get into troubles, where limiting functions can have a strong impact on the outcome of the simulations Klee et al (2017). (Young & Clarke 2015) also observed fragmentation above the critical β crit in their simulations.…”