The Maxey-Riley equations (MRE) describe the motion of a finite-sized, spherical particle in a fluid. Because of wake effects, the force acting on a particle depends on its past trajectory. This is modelled by an integral term in the MRE, also called Basset force, that makes its numerical solution challenging and memory intensive. A recent approach proposed by Prasath, Vasan and Govindarajan exploits connections between the integral term and fractional derivatives to reformulate the MRE as a time-dependent partial differential equation on a semi-infinite pseudo-space. They also propose a numerical algorithm based on polynomial expansions. This paper develops a numerical approach based on finite difference instead, by adopting techniques by Koleva and Fazio and Janelli to cope with the issues of having an unbounded spatial domain. We compare convergence order and computational efficiency for particles of varying size and density of the polynomial expansion by Prasath et al., our finite difference schemes and a direct integrator for the MRE based on multi-step methods proposed by Daitche.

翻译：Maxey-Riley方程描述了有限尺寸球形颗粒在流体中的运动。由于尾流效应，颗粒所受作用力取决于其历史运动轨迹。这一特性通过方程中的积分项（即Basset力）进行建模，导致数值求解困难且计算内存消耗巨大。Prasath、Vasan与Govindarajan近期提出的方法利用积分项与分数阶导数之间的关联，将Maxey-Riley方程重新表述为关于半无限伪空间的时间依赖偏微分方程，并提出了基于多项式展开的数值算法。本文采用Koleva、Fazio及Janelli处理无界空间域问题的技术，转而发展基于有限差分的数值方法。通过比较不同粒径与密度颗粒情况下的收敛阶及计算效率，本文分析了Prasath等多项式展开方法、本文提出的有限差分格式，以及Daitche基于多步法提出的Maxey-Riley方程直接积分器的性能差异。