An efficient high order numerical scheme for the time-fractional diffusion equation with uniform accuracy

The construction of efficient numerical schemes with uniform convergence order for time-fractional diffusion equations (TFDEs) is an important research problem. We are committed to study an efficient uniform accuracy scheme for TFDEs. Firstly, we use the piecewise quadratic interpolation to construct an efficient uniform accuracy scheme for the fractional derivative of time. And the local truncation error of the efficient scheme is also given. Secondly, the full discrete numerical scheme for TFDEs is given by combing the spatial center second order scheme and the above efficient time scheme. Thirdly, the efficient scheme's stability and error estimates are strictly theoretical analysis to obtain that the unconditionally stable scheme is $3-\beta$ convergence order with uniform accuracy in time. Finally, some numerical examples are applied to show that the proposed scheme is an efficient unconditionally stable scheme.