Explicit Runge-Kutta generalized Exponential Time-Differencing method to solve full Maxey-Riley equation with the history force

Fractional differential equations (FDE) are numerically expensive to solve because fractional derivatives are nonlocal quantites and the associated linear integrator (the function that solves the linear part of the FDE exactly) can lack semigroup property leading to increasing memory storage cost. The Maxey-Riley (MR) equation that models the advection of a particle in viscous flow contains the Basset-Boussinesq history force which is a half-derivative, and thus inherits all the numerical challenges of solving an FDE. We have developed an explicit time-integrator, inspired by generalizing exponential time differencing method, to solve the MR equation (and a more general subclass of FDEs) that incurs a fixed-in-time computational and memory cost with tunable accuracy. We show that the lack of semigroup property of the linear integrator can be dealt with by suitably embedding the MR equation into a larger Markovian system. The resulting extended system can subsequently be evolved locally by suitably adapting the standard ideas from explicit time-integrator methods. We demonstrate the computational performance and accuracy of our method for the exemplar case of a particle advected by the MR equation in Couette flow.