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 α ∈ (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 σ 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. [ABSTRACT FROM AUTHOR]