Local Discontinuous Galerkin Method Coupled with Nonuniform Time Discretizations for Solving the Time-Fractional Allen-Cahn Equation

This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order alpha is an element of(0,1). Considering the weak singularity of the solution u(x,t) at the starting time, i.e., its first and/or second derivatives with respect to time blowing-up as t -> 0(+) albeit the function itself being right continuous at t=0, two well-known difference formulas, including the nonuniform L1 formula and the nonuniform L2-1(sigma) formula, which are used to approximate the Caputo time-fractional derivative, respectively, and the local discontinuous Galerkin (LDG) method is applied to discretize the spatial derivative. With the help of discrete fractional Gronwall-type inequalities, the stability and optimal error estimates of the fully discrete numerical schemes are demonstrated. Numerical experiments are presented to validate the theoretical results.