A high-order compact difference scheme on graded mesh for time-fractional Burgers' equation

In this work, we studied a compact difference scheme for solving the time-fractional Burgers' equation. Using the L2-1 sigma formula on the graded mesh to approximate the fractional derivative in temporal direction and a novel nonlinear fourth-order compact difference operator discrete the spatial nonlinear term, we proposed a discrete scheme that could handle the initial weak singularity of the solution as well as preserve high accuracy in the space direction. We first prove the existence and boundedness of the numerical solution. Then the stability and the convergence of the scheme are rigorously analyzed using the energy method. The theoretical result shows that with an appropriate choice of graded mesh, the proposed scheme could reach second-order accuracy in time and fourth-order accuracy in space. Numerical experiments are also carried out to verify the theoretical results.