  It is well-known that the Allen-Cahn equation not only satisfies the energy
dissipation law but also possesses the maximum bound principle (MBP) in the
sense that the absolute value of its solution is pointwise bounded for all time
by some specific constant under appropriate initial/boundary conditions. In
recent years, the scalar auxiliary variable (SAV) method and many of its
variants have attracted much attention in numerical solution for gradient flow
problems due to their inherent advantage of preserving certain discrete
analogues of the energy dissipation law. However, existing SAV schemes usually
fail to preserve the MBP when applied to the Allen-Cahn equation. In this
paper, we develop and analyze new first- and second-order stabilized
exponential-SAV schemes for a class of Allen-Cahn type equations, which are
shown to simultaneously preserve the energy dissipation law and MBP in discrete
settings. In addition, optimal error estimates for the numerical solutions are
rigorously obtained for both schemes. Extensive numerical tests and comparisons
are also conducted to demonstrate the performance of the proposed schemes.
