  We present a set of linear, second order, unconditionally energy stable
schemes for the Allen-Cahn equation with nonlocal constraints that preserves
the total volume of each phase in a binary material system. The energy
quadratization strategy is employed to derive the energy stable semi-discrete
numerical algorithms in time. Solvability conditions are then established for
the linear systems resulting from the semi-discrete, linear schemes. The fully
discrete schemes are obtained afterwards by applying second order finite
difference methods on cell-centered grids in space. The performance of the
schemes are assessed against two benchmark numerical examples, in which
dynamics obtained using the volumepreserving Allen-Cahn equations with nonlocal
constraints is compared with those obtained using the classical Allen-Cahn as
well as the Cahn-Hilliard model, respectively, demonstrating slower dynamics
when volume constraints are imposed as well as their usefulness as alternatives
to the Cahn-Hilliard equation in describing phase evolutionary dynamics for
immiscible material systems while preserving the phase volumes. Some
performance enhancing, practical implementation methods for the linear energy
stable schemes are discussed in the end.
