A supper-convergent approximation on temporal nonuniform meshes for the linear combination of multiple Caputo time-fractional derivative operators with different orders in (0, 1) is constructed and analyzed. This approximation is based on piecewise quadratic and linear interpolation polynomials at a certain super-convergence point, so it is called the \(L2\) - \(1_{\sigma}\) approximation of the multi-term time-fractional derivative operator on temporal nonuniform meshes. A monotone iterative algorithm with second-order convergence rate is developed for the first time to obtain the supper-convergence point. Combining the \(L2\) - \(1_{\sigma}\) approximation with the fourth-order compact exponential difference approximation of the spatial differential operator, a high-order \(L2\) - \(1_{\sigma}\) compact exponential difference method is proposed for a class of multi-term time-fractional convection-reaction-diffusion equations. The unconditional stability and convergence of the method are proved by using the discrete energy method and appropriately decomposing the asymmetric coefficient matrix of the discrete scheme. The proposed method has spatial fourth-order convergence, and for initially weakly singular solutions, it can achieve temporal optimal second-order convergence when the temporal mesh parameter is selected properly. Numerical results confirm the theoretical analysis results and demonstrate the applicability of the proposed method to convection-dominated problems.
