Explicit integrators for nonlocal equations: The case of the Maxey-Riley-Gatignol equation

Abstract

The Maxey-Riley-Gatignol (MRG) equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, is an integro-differential equation with a memory term and its solution lacks a well-defined Taylor series at t = 0 t=0 . In particulate flows, one often seeks trajectories of millions of particles simultaneously, and the numerical solution to the MRG equation for each particle becomes prohibitively expensive due to its ever-rising memory costs. In this paper, we present an explicit numerical integrator for the MRG equation that inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition. The integrator is based on a Markovian embedding of the MRG equation. The integrator and the embedding are consequences of a spectral representation of the solution to the linear MRG equation. We exploit these to extend the work of Cox and Matthews [J. Comput. Phys. 176 (2002), 430–455] and derive Runge-Kutta type iterative schemes of differing orders for the MRG equation. Our approach may be generalized to a large class of systems with memory effects.