The Variable-Step L1 Scheme Preserving a Compatible Energy Law for Time-Fractional Allen-Cahn Equation

In this work, we revisit the adaptive L1 time-stepping scheme for solving time-fractional Allen-Cahn equation in Caputo's form. The implicit is shown to preserve a variational energy dissipation law on arbitrary nonuniform time meshes by using recent discrete analysis tools, i.e., orthogonal convolution kernels and complementary kernels. Then embedding techniques fractional Gr\"onwall inequality were applied establish an $L^2$ norm error estimate meshes. An strategy according dynamical feature of system presented capture multi-scale behaviors improve computational performance.