We consider the efficient numerical solution of coupled dynamical systems, consisting of a low dimensional nonlinear part and a high dimensional linear time invariant part, e.g., stemming from spatial discretization of an underlying partial differential equation. The linear subsystem can be eliminated in frequency domain and for the numerical solution of the resulting integro-differential algebraic equations, we propose a combination of Runge–Kutta or multistep time stepping methods with appropriate convolution quadrature to handle the integral terms. The resulting methods are shown to be algebraically equivalent to a Runge–Kutta or multistep solution of the coupled system and thus automatically inherit the corresponding stability and accuracy properties. After a computationally expensive pre-processing step, the online simulation can, however, be performed at essentially the same cost as solving only the low dimensional nonlinear subsystem. The proposed method is, therefore, particularly attractive, if repeated simulation of the coupled dynamical system is required.

我们考虑耦合动力学系统的有效数值解，该动力学解包括低维非线性部分和高维线性时不变部分，例如，源于基础偏微分方程的空间离散化。线性子系统可以在频域中消除，并且对于所得积分微分代数方程的数值解，我们建议将Runge–Kutta或多步时间步长方法与适当的卷积正交相结合，以处理积分项。结果表明，所得方法与耦合系统的Runge-Kutta或多步解决方案在数学上等效，因此可以自动继承相应的稳定性和准确性。但是，在经过计算昂贵的预处理步骤后，在线仿真可以 与仅解决低维非线性子系统的成本基本相同。因此，如果需要对耦合动力系统进行重复仿真，则所提出的方法特别有吸引力。