Pointwise-in-time error analysis of the corrected L1 scheme for a time-fractional sine-Gordon equation

In this work, the time-fractional sine-Gordon equation with Neumann boundary conditions is considered, where the solutions exhibit typical weak singularities at initial time. By introducing an intermediate variable, the original problem can be equivalently written as a low-order coupled system. Utilizing the nonuniform corrected L1 scheme in time and the finite difference scheme in space, a fully discrete scheme is constructed for the coupled system. Furthermore, the stability of the proposed scheme is rigorously established. Meanwhile, a sharp pointwise-in-time error analysis is developed. In particular, the deriving convergent result implies that the error away from the initial time reaches the optimal convergence rate of $2-\alpha /2$ by merely taking the grading parameter $r=1$ for any $1<\alpha <2$. Finally, numerical results are provided to verify that the proposed scheme achieves optimal convergence rates in both temporal and spatial directions.