A Unified Fast Memory-Saving Time-Stepping Method for Fractional Operators and Its Applications

Time-dependent fractional partial differential equations typically require huge amounts of memory and computational time, especially for long-time integra-tion, which taxes computational resources heavily for high-dimensional problems. Here, we first analyze existing numerical methods of sum-of-exponentials for ap-proximating the kernel function in constant-order fractional operators, and iden-tify the current pitfalls of such methods. In order to overcome the pitfalls, an im-proved sum-of-exponentials is developed and verified. We also present several sum-of-exponentials for the approximation of the kernel function in variable-order frac-tional operators. Subsequently, based on the sum-of-exponentials, we propose a uni-fied framework for fast time-stepping methods for fractional integral and derivative operators of constant and variable orders. We test the fast method based on several benchmark problems, including fractional initial value problems, the time-fractional Allen-Cahn equation in two and three spatial dimensions, and the Schro spacing diaeresis dinger equa-tion with nonreflecting boundary conditions, demonstrating the efficiency and robustness of the proposed method. The results show that the present fast method significantly reduces the storage and computational cost especially for long-time integration problems.