A standard approach to solve ordinary differential equations, when they describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such schemes, however, are not applicable to the large class of equations which do not constitute dynamical systems. In several physical systems, we encounter integro-differential equations with memory terms where the time derivative of a state variable at a given time depends on all past states of the system. Secondly, there are equations whose solutions do not have well-defined Taylor series expansion. The Maxey-Riley-Gatignol equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, displays both challenges. We use it as a test bed to address the questions we raise, but our method may be applied to all equations of this class. We show that the Maxey-Riley-Gatignol equation can be embedded into an extended Markovian system which is constructed by introducing a new dynamical co-evolving state variable that encodes memory of past states. We develop a Runge-Kutta algorithm for the resultant Markovian system. The form of the kernels involved in deriving the Runge-Kutta scheme necessitates the use of an expansion in powers of $t^{1/2}$. Our approach naturally 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.

翻译：求解常微分方程（当它们描述动力系统时）的标准方法是采用龙格-库塔或其相关格式。然而，这些格式不适用于不构成动力系统的大类方程。在若干物理系统中，我们遇到带有记忆项的积分-微分方程，其中状态变量在给定时刻的时间导数依赖于系统所有过去的状态。其次，存在解不具有明确定义的泰勒级数展开的方程。描述惯性粒子在非均匀非定常流中动力学的Maxey-Riley-Gatignol方程同时展现了这两个挑战。我们将其作为验证所提出问题的测试基准，但我们的方法可应用于此类所有方程。我们证明Maxey-Riley-Gatignol方程可嵌入一个扩展的马尔可夫系统，该系统的构建通过引入一个编码过去状态记忆的新的动态协进化状态变量实现。我们为由此产生的马尔可夫系统开发了一种龙格-库塔算法。推导龙格-库塔格式所涉及的核函数形式决定了必须使用$t^{1/2}$的幂展开。我们的方法自然继承了标准时间积分器的优势，即恒定存储成本、运算开销随模拟时间线性增长，以及能够以最终状态作为新初始条件重新启动模拟。