论文标题
为普通微分方程保留Lyapunov函数的数值方法
Numerical methods that preserve a Lyapunov function for Ordinary Differential Equations
论文作者
论文摘要
论文研究了保留动力学系统的lyapunov函数的数值方法，即能量降低的数值近似值，就像在原始微分方程中一样。以此目的，实现了一种离散的梯度方法，以用于数值集成普通微分方程系统。原则上，此过程产生一级方法，但是分析为高阶方法设计铺平了道路。作为一个很好的例子，考虑到在这种情况下，保留Lyapunov函数比特定轨迹的准确性更为重要，将提出的方法应用于行李方程。结果通过数值实验验证，其中将离散的梯度方法与标准runge-kutta方法进行了比较。正如该理论所预测的那样，离散的梯度方法保留了Lyapunov函数，而常规方法则无法做到，因为要进行周期性解决方案或能量不会降低。此外，就计算成本而言，当这些方法确实保留了Lyapunov函数时，离散的梯度方法优于常规方案，因此提出的方法是有希望的。
The paper studies numerical methods that preserve a Lyapunov function of a dynamical system, i.e. numerical approximations whose energy decreases, just like in the original differential equation. With this aim, a discrete gradient method is implemented for numerical integration of a system of ordinary differential equations. In principle, this procedure yields first order methods, but the analysis paves the way to the design of higher-order methods. As a case in point, the proposed method is applied to the Duffing equation without external forcing, considering that in this case, preserving the Lyapunov function is more important than accuracy of particular trajectories. Results are validated by means of numerical experiments, where the discrete gradient method is compared to standard Runge-Kutta methods. As predicted by the theory, discrete gradient methods preserve the Lyapunov function, whereas conventional methods fail to do so, since either periodic solutions appear or the energy does not decrease. Besides, the discrete gradient method outperforms conventional schemes when these do preserve the Lyapunov function, in terms of computational cost, thus the proposed method is promising.
