  In this work, we develop implicit rank-adaptive schemes for time-dependent
matrix differential equations. The dynamic low rank approximation (DLRA) is a
well-known technique to capture the dynamic low rank structure based on
Dirac-Frenkel time-dependent variational principle. In recent years, it has
attracted a lot of attention due to its wide applicability. Our schemes are
inspired by the three-step procedure used in the rank adaptive version of the
unconventional robust integrator (the so called BUG integrator) for DLRA.
First, a prediction (basis update) step is made computing the approximate
column and row spaces at the next time level. Second, a Galerkin evolution step
is invoked using a base implicit solve for the small core matrix. Finally, a
truncation is made according to a prescribed error threshold. Since the DLRA is
evolving the differential equation projected on to the tangent space of the low
rank manifold, the error estimate of the BUG integrator contains the tangent
projection (modeling) error which cannot be easily controlled by mesh
refinement. This can cause convergence issue for equations with cross terms.
  To address this issue, we propose a simple modification, consisting of
merging the row and column spaces from the explicit step truncation method
together with the BUG spaces in the prediction step. In addition, we propose an
adaptive strategy where the BUG spaces are only computed if the residual for
the solution obtained from the prediction space by explicit step truncation
method, is too large. We prove stability and estimate the local truncation
error of the schemes under assumptions. We benchmark the schemes in several
tests, such as anisotropic diffusion, solid body rotation and the combination
of the two, to show robust convergence properties.
