论文标题
动力学系统的自适应光谱Koopman方法
The Adaptive Spectral Koopman Method for Dynamical Systems
论文作者
论文摘要
动力学系统在力学，电气工程，化学等方面具有广泛的应用。在这项工作中，我们提出了自适应光谱Koopman（询问）方法来解决非线性自主动力学系统。这种新颖的数值方法利用光谱 - 同盟（即伪谱）方法和库普曼操作员的性能来获得动力学系统的解决方案。具体而言，该解决方案表示为Koopman操作员特征函数和特征值的乘法的线性组合，并且这些特征值通过光谱方法近似。与传统的时间演化算法（例如Euler的方案和Runge-Kutta方案）不同，ASK是无网格的，因此在评估解决方案时更加灵活。数值实验表明，要求解决一，二维和三维动力系统的高精度。
Dynamical systems have a wide range of applications in mechanics, electrical engineering, chemistry, and so on. In this work, we propose the adaptive spectral Koopman (ASK) method to solve nonlinear autonomous dynamical systems. This novel numerical method leverages the spectral-collocation (i.e., pseudo-spectral) method and properties of the Koopman operator to obtain the solution of a dynamical system. Specifically, this solution is represented as a linear combination of the multiplication of Koopman operator's eigenfunctions and eigenvalues, and these eigenpairs are approximated by the spectral method. Unlike conventional time evolution algorithms such as Euler's scheme and the Runge-Kutta scheme, ASK is mesh-free, and hence is more flexible when evaluating the solution. Numerical experiments demonstrate high accuracy of ASK for solving one-, two- and three-dimensional dynamical systems.
