  Dynamical low-rank approximation has become a valuable tool to perform an
on-the-fly model order reduction for prohibitively large matrix differential
equations. A core ingredient is the construction of integrators that are robust
to the presence of small singular values and the resulting large time
derivatives of the orthogonal factors in the low-rank matrix representation.
Recently, the robust basis-update & Galerkin (BUG) class of integrators has
been introduced. These methods require no steps that evolve the solution
backward in time, often have favourable structure-preserving properties, and
allow for parallel time-updates of the low-rank factors. The BUG framework is
flexible enough to allow for adaptations to these and further requirements.
However, the BUG methods presented so far have only first-order robust error
bounds. This work proposes a second-order BUG integrator for dynamical low-rank
approximation based on the midpoint rule. The integrator first performs a
half-step with a first-order BUG integrator, followed by a Galerkin update with
a suitably augmented basis. We prove a robust second-order error bound which in
addition shows an improved dependence on the normal component of the vector
field. These rigorous results are illustrated and complemented by a number of
numerical experiments.
