Numerical energy dissipation for time fractional volume-conserved Allen–Cahn model based on the ESAV and R-ESAV approaches

In this study, we consider volume-conserved numerical schemes for the volume-conserved time fractional Allen–Cahn equation. We start with the L1 scheme based on a modified exponential scalar auxiliary variable (ESAV) approach for handling the nonlinear potential term. Then we introduce the fast L1 scheme to significantly reduce the CPU time and to make the long-term computation possible. Further to reduce the error between the discrete energy and the original energy of the ESAV approach, we introduce the fast L1 scheme based on a relaxed modified ESAV (R-ESAV) idea. We rigorously show the discrete energy dissipation properties for the above three schemes, including the discrete energy boundedness law, the discrete fractional energy dissipation law, and the discrete weighted energy dissipation law. Theoretical findings are validated by a few numerical experiments.