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In this paper, we propose a fast second-order approximation to thevariable-order (VO) Caputo fractional derivative, which is developed based on$L2$-$1_\sigma$ formula and the exponential-sum-approximation technique. Thefast evaluation method can achieve the second-order accuracy and further reducethe computational cost and the acting memory for the VO Caputo fractionalderivative. This fast algorithm is applied to construct a relevant fasttemporal 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 similarproperties of the coefficients as those of the well-studied $L2$-$1_\sigma$scheme. Therefore, $FL2$-$1_{\sigma}$ scheme is strictly proved to beunconditionally stable and convergent. A sharp decrease in the computationalcost and the acting memory is shown in the numerical examples to demonstratethe efficiency of the proposed method.

Fast second-order evaluation for variable-order Caputo fractional derivative with applications to fractional sub-diffusion equations