A Second-Order in Time and Energy-Dissipative Scheme for Time-Fractional Navier-Stokes Equations

Abstract

This paper proposes an energy-dissipative scheme for solving two- and three-dimensional time-fractional Navier-Stokes equations. The numerical scheme is constructed, using nonuniform $L2−1σ$ approximation in the temporal direction and the Fourier spectral method in the spatial direction. It is shown that the numerical scheme can keep discrete energy stable and the numerical solutions are uniformly bounded without any restriction on step sizes. Error estimates of the fully-discrete scheme are presented. Moreover, a fast algorithm is applied to accelerate the computation. Numerical results in long time intervals are presented to confirm the effectiveness and high efficiency of the scheme.