A second-order energy stable and nonuniform time-stepping scheme for time fractional Burgers' equation

Abstract

In this paper, an effective finite difference scheme of high order accuracy is proposed for the nonlinear time fractional Burgers' equation. Specifically, we apply the Alikhanov's scheme on graded mesh in the temporal direction and a novel fourth-order compact scheme in the spatial discretization. The proposed scheme resolves initial weak singularity of the solution and preserves high resolution in the space direction. It is rigorously proved that the finite difference scheme is uniquely solvable, discrete variational energy dissipation law and unconditionally stable and convergent in sense of discrete L 2 -norm. With appropriate choice of the grading parameter, the convergence accuracy is min ⁡ { r α , 2 } order in time and fourth order in space, where r is the mesh grading. In the numerical implementation procedure, fast convolution technique and adaptive time-stepping strategy are adopted to accelerate the presented solver and to capture evolution of the solution. Numerical experiments are carried out to verify the validity and effectiveness of the proposed scheme for solving nonlinear time-fractional Burgers' equation.