  A framework for performant Brownian Dynamics (BD) many-body simulations with
adaptive timestepping is presented. Contrary to the Euler-Maruyama scheme in
common non-adaptive BD, we employ an embedded Heun-Euler integrator for the
propagation of the overdamped coupled Langevin equations of motion. This
enables the derivation of a local error estimate and the formulation of
criteria for the acceptance or rejection of trial steps and for the control of
optimal stepsize. Introducing erroneous bias in the random forces is avoided by
Rejection Sampling with Memory (RSwM) due to Rackauckas and Nie, which makes
use of the Brownian bridge theorem and guarantees the correct generation of a
specified random process even when rejecting trial steps. For test cases of
Lennard-Jones fluids in bulk and in confinement, it is shown that adaptive BD
solves performance and stability issues of conventional BD, already
outperforming the latter even in standard situations. We expect this novel
computational approach to BD to be especially helpful in long-time simulations
of complex systems, e.g. in non-equilibrium, where concurrent slow and fast
processes occur.
