  In this paper, we propose a fast second-order approximation to the
variable-order (VO) Caputo fractional derivative, which is developed based on
$L2$-$1_\sigma$ formula and the exponential-sum-approximation technique. The
fast evaluation method can achieve the second-order accuracy and further reduce
the computational cost and the acting memory for the VO Caputo fractional
derivative. This fast algorithm is applied to construct a relevant fast
temporal second-order and spatial fourth-order scheme ($FL2$-$1_{\sigma}$
scheme) for the multi-dimensional VO time-fractional sub-diffusion equations.
Theoretically, $FL2$-$1_{\sigma}$ scheme is proved to fulfill the similar
properties of the coefficients as those of the well-studied $L2$-$1_\sigma$
scheme. Therefore, $FL2$-$1_{\sigma}$ scheme is strictly proved to be
unconditionally stable and convergent. A sharp decrease in the computational
cost and the acting memory is shown in the numerical examples to demonstrate
the efficiency of the proposed method.
