Novel conservative methods for adaptive force softening in collisionless and multispecies N-body simulations

Modelling self-gravity of collisionless fluids (e.g. ensembles of dark matter, stars, black holes, dust, and planetary bodies) in simulations is challenging and requires some force softening. It is often desirable to allow softenings to evolve adaptively, in any high-dynamic range simulation, but this poses unique challenges of consistency, conservation, and accuracy, especially in multiphysics simulations where species with different 'softening laws' may interact. We therefore derive a generalized form of the energy-and-momentum conserving gravitational equations of motion, applicable to arbitrary rules used to determine the force softening, together with consistent associated time-step criteria, interaction terms between species with different softening laws, and arbitrary maximum/minimum softenings. We also derive new methods to maintain better accuracy and conservation when symmetrizing forces between particles. We review and extend previously discussed adaptive softening schemes based on the local neighbour particle density, and present several new schemes for scaling the softening with properties of the gravitational field, i.e. the potential or acceleration or tidal tensor. We show that the 'tidal softening' scheme not only represents a physically motivated, translation and Galilean invariant and equivalence-principle respecting (and therefore conservative) method but also imposes negligible time-step or other computational penalties, ensuring that pairwise two-body scattering is small compared to smooth background forces and can resolve outstanding challenges in properly capturing tidal disruption of substructures (minimizing artificial destruction) while also avoiding excessive N-body heating. We make all of this public in the GIZMO code.