Grain growth for astrophysics with discontinuous Galerkin schemes

Abstract

Depending on their sizes, dust grains store more or less charges, catalyse more or less chemical reactions, intercept more or less photons and stick more or less efficiently to form embryos of planets. Hence the need for an accurate treatment of dust coagulation and fragmentation in numerical modelling. However, existing algorithms for solving the coagulation equation are over-diffusive in the conditions of 3D simulations. We address this challenge by developing a high-order solver based on the Discontinuous Galerkin method. This algorithm conserves mass to machine precision and allows to compute accurately the growth of dust grains over several orders of magnitude in size with a very limited number of dust bins.

On the Courant-Friedrichs-Lewy condition for numerical solvers of the coagulation equation

“…3). We validate these findings by testing this condition back in the mass space with the solver of Lombart & Laibe (2021). In this study, we focus on constant and additive kernels, since they can be associated with analytic solutions that are the most relevant for planet formation.…”

“…10 in the mass space. We therefore solve the Smoluchowski equation with the algorithm of Lombart & Laibe (2021). We use 9 orders of magnitude in mass and n = 15 log-spaced bins to mimic the challenging integration conditions encountered in practice.…”

“…However, this task was long thought to be computationally prohibitive, since no hydrodynamical code could handle the large number of dust bins required to solve for the coagulation equation without over-diffusion. Recently, Lombart & Laibe (2021) showed that over-diffusivity at small bin numbers could be overcome by the mean of a Discontinuous Galerkine algorithm of high spatial order (Liu et al 2019). Still, to maintain practical performance, the coagulation solver should not be called too often per hydrodynamical time step (Dr ążkowska et al 2014(Dr ążkowska et al , 2019.…”

Existence of mass conserving solution for the coagulation–fragmentation equation with singular kernel

“…Over the years many researchers have studied the nonlinear behaviour of colisional fragmentation model, e.g., scaling solutions [14,18,23], shattering behaviour [1,25], existenceuniqueness and well-posedness of weak solution [26], Monte Carlo (Direct simulation) algorithm [27], discontinuous Galerkin scheme [28], etc. However, to the best of authors knowledge the existence-uniqueness result of a global mass preserving continuous solution for singular kernel collision rate is not studied yet.…”

A Discontinuous Galerkin Solver in the FLASH Multi-Physics Framework

“…However, their application in astrophysics for hypersonic flows is still in an early stage. DG methods with focus on astrophysical fluid dynamics and related applications are for instance presented in Schaal et al (2015); Guillet et al (2019); Bauer et al (2016); Kidder et al (2017); Lombart & Laibe (2020).…”