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It is well-known that the Allen-Cahn equation not only satisfies the energydissipation law but also possesses the maximum bound principle (MBP) in thesense that the absolute value of its solution is pointwise bounded for all timeby some specific constant under appropriate initial/boundary conditions. Inrecent years, the scalar auxiliary variable (SAV) method and many of itsvariants have attracted much attention in numerical solution for gradient flowproblems due to their inherent advantage of preserving certain discreteanalogues of the energy dissipation law. However, existing SAV schemes usuallyfail to preserve the MBP when applied to the Allen-Cahn equation. In thispaper, we develop and analyze new first- and second-order stabilizedexponential-SAV schemes for a class of Allen-Cahn type equations, which areshown to simultaneously preserve the energy dissipation law and MBP in discretesettings. In addition, optimal error estimates for the numerical solutions arerigorously obtained for both schemes. Extensive numerical tests and comparisonsare also conducted to demonstrate the performance of the proposed schemes.

Stabilized exponential-SAV schemes preserving energy dissipation law and maximum bound principle for the Allen-Cahn type equations