标题： 保持对称性的数值格式 显示英文标题

标题： Symmetry-preserving numerical schemes

摘要： 在这些讲座中，我们回顾了两种构建保持微分方程对称性的有限差分数值格式的程序。 第一种方法基于李的无穷小对称生成元，而第二种方法使用了等变移动框架的新理论。 讨论并举例说明了这两种技术的优势，包括施瓦茨微分方程、科尔特韦格-德弗里斯方程和伯格斯方程。 展示了数值模拟，并介绍了获得更好不变数值格式的创新技术。 在这些笔记的最后指出了新的研究方向和开放问题。 显示英文摘要

摘要： In these lectures we review two procedures for constructing finite difference numerical schemes that preserve symmetries of differential equations. The first approach is based on Lie's infinitesimal symmetry generators, while the second method uses the novel theory of equivariant moving frames. The advantages of both techniques are discussed and illustrated with the Schwarzian differential equation, the Korteweg-de Vries equation and Burgers' equation. Numerical simulations are presented and innovative techniques for obtaining better invariant numerical schemes are introduced. New research directions and open problems are indicated at the end of these notes.