In this paper, we employ two modified L2-1 \(_{\sigma}\) schemes coupled with a spectral method to numerically solve a variable-coefficient time-space fractional diffusion equation, where the fractional Laplacian is defined in its pseudo-differential definition. In the time direction, an improved L2-1 \(_{\sigma}\) scheme and a fast L2-1 \(_{\sigma}\) scheme are used to treat the Caputo fractional derivative. Both schemes are formulated based on hybrid time grids that unify graded meshes and uniform meshes, which enable them to efficiently improve the computational accuracy when approximating the exact solution with initial weak singularity. Moreover, the former scheme is derived by optimizing the discrete coefficients of the direct L2-1 \(_{\sigma}\) scheme, and it retains numerical accuracy even for small time-fractional exponents if the gradient index is relatively large. Meanwhile, the latter scheme is derived based on the L2-1 \(_{\sigma}\) scheme and the sum-of-exponents technique, and it enables one to reduce the computational cost significantly. In the space direction, an effective and simple spectral method based on the semi-discrete Fourier transform is employed to cope with the fractional Laplacian, and it yields spectral accuracy if the solution is sufficiently smooth. Further, rigorous theoretical analyses regarding the stability and convergence of the corresponding fully discrete schemes are conducted. Finally, numerical experiments are performed to validate the effectiveness of the proposed schemes.
