Fast Evaluation of the Caputo Fractional Derivative and Its Applications to Fractional Diffusion Equations: A Second-Order Scheme (Communications in Computational Physics, 2017)
The fractional derivatives include nonlocal information and thus their calculationrequires huge storage and computational cost for long time simulations. Wepresent an efficient and high-order accurate numerical formula to speed up the evaluationof the Caputo fractional derivative based on the L2-1σ formula proposed in[A. Alikhanov, J. Comput. Phys., 280 (2015), pp. 424-438], and employing the sum-of-exponentialsapproximation to the kernel function appeared in the Caputo fractionalderivative. Both theoretically and numerically, we prove that while applied to solvingtime fractional diffusion equations, our scheme not only has unconditional stabilityand high accuracy but also reduces the storage and computational cost.