  In this report, we propose a new adaptive time filter algorithm for the
unsteady Stokes/Darcy model. First we present a first order ${\theta}$-scheme
with the variable time step which is one parameter family of Linear Multi-step
methods and use a time filter algorithm to increase the convergence order to
second order with almost no increasing the amount of computation. Furthermore,
we construct coupled and decoupled adaptive algorithms. Then we analyze
stabilities and the second-order accuracy of variable time-stepping algorithm
for Linear Multi-step methods plus time filter, respectively. Finally, we use
two numerical experiments to verify theoretical results including
effectiveness, convergence and efficiency with adaptive method.

Numerical Analysis;Numerical Analysis