  In the report, we propose a family of variable time-stepping ensemble
algorithms for solving multiple incompressible Navier-Stokes equations (NSE) at
one pass. The one-leg, two-step methods designed by Dahlquist, Liniger, and
Nevanlinna (henceforth the DLN method) are non-linearly stable and second-order
accurate under arbitrary time grids. We design the family of variable
time-stepping DLN-Ensemble algorithms for multiple systems of NSE and prove
that its numerical solutions are stable and second-order accurate in velocity
under moderate time-step restrictions. Meanwhile, the family of algorithms can
be equivalently implemented by a simple refactorization process: adding time
filters on the backward Euler ensemble algorithm. In practice, we raise one
time adaptive mechanism (based on the local truncation error criterion) for the
family of DLN-Ensemble algorithms to balance accuracy and computational costs.
Several numerical tests are to support the main conclusions of the report. The
constant step test confirms the second-order convergence and time efficiency.
The variable step test verifies the stability of the numerical solutions and
the time efficiency of the adaptive mechanism.
