  We present error estimates for four unconditionally energy stable numerical
schemes developed for solving Allen-Cahn equations with nonlocal constraints.
The schemes are linear and second order in time and space, designed based on
the energy quadratization (EQ) or the scalar auxiliary variable (SAV) method,
respectively. In addition to the rigorous error estimates for each scheme, we
also show that the linear systems resulting from the energy stable numerical
schemes are all uniquely solvable. Then, we present some numerical experiments
to show the accuracy of the schemes, their volume-preserving as well as energy
dissipation properties in a drop merging simulation.
