隐式常微分方程的数值分析与可视化
On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations
摘要：
我们讨论了微分方程的几何理论如何用于隐式常微分方程的数值积分和可视化，特别是在方程的奇异点周围。Vessiot理论自动将隐式微分方程转换为流形上的向量场分布，从而将其分析简化为动力系统理论中的标准问题，如向量场的积分和不变流形的确定。对于低维的情况下，我们适应流线算法的Jobard和Lefer的2.5和3维。讨论了在Matlab中的具体实现，并给出了具体的例子。
英文摘要：
We discuss how the geometric theory of differential equations can be used for the numerical integration and visualisation of implicit ordinary differential equations, in particular around singularities of the equation. The Vessiot theory automatically transforms an implicit differential equation into a vector field distribution on a manifold and thus reduces its analysis to standard problems in dynamical systems theory like the integration of a vector field and the determination of invariant manifolds. For the visualisation of low-dimensional situations we adapt the streamlines algorithm of Jobard and Lefer to 2.5 and 3 dimensions. A concrete implementation in Matlab is discussed and some concrete examples are presented.
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