A class of efficient time-stepping methods for multi-term time-fractional reaction-diffusion-wave equations

Abstract A family of novel time-stepping methods for the fractional calculus operators is presented with a shifted parameter. The truncation error with second-order accuracy is proved under the framework of the shifted convolution quadrature. To improve the efficiency, two aspects are considered, that i) a fast algorithm is developed to reduce the computation complexity from O ( N t 2 ) to O ( N t log ⁡ N t ) and the memory requirement from O ( N t ) to O ( log ⁡ N t ) , where N t denotes the number of successive time steps, and ii) correction terms are added to deal with the initial singularity of the solution. The stability analysis and error estimates are provided in detail where in temporal direction the novel time-stepping methods are applied and the spatial variable is discretized by the finite element method. Numerical results for d-dimensional examples ( d = 1 , 2 , 3 ) confirm our theoretical conclusions and the efficiency of the fast algorithm.

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