Generalized Exponential Time Differencing Schemes for Stiff Fractional Systems with Nonsmooth Source Term

Many processes in science and engineering are described by fractional systems which may in general be stiff and involve a nonsmooth source term. In this paper, we develop robust first, second, and third order accurate exponential time differencing schemes for solving such systems. Rather than imposing regularity requirements on the solution to account for the singularity caused by the fractional derivative, we only consider regularity requirements on the source term for preserving the optimal order of accuracy of the proposed schemes. Optimal convergence rates are proved for both smooth and nonsmooth source terms using uniform and graded meshes, respectively. For efficient implementation, high-order global Pade approximations together with their fractional decompositions are developed for Mittag-Leffler functions. We present numerical experiments involving a typical stiff system, a fractional two-compartment pharmacokinetics model, a two-term fractional Kelvin-Viogt model of viscoelasticity, and a large system obtained by spatial discretization of a sub-diffusion problem. Demonstrations of the efficiency of the rational approximation implementation technique and the newly constructed high-order schemes are provided.