A note on continuous-stage Runge-Kutta methods
A note on continuous-stage Runge-Kutta methods
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DOI：
10.1016/j.amc.2018.07.044
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发表时间：
2018-04
期刊：
影响因子：
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通讯作者：
Wensheng Tang
中科院分区：
文献类型：
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作者：
Wensheng Tang
本文给出了求解一阶常微分方程初值问题的连续级龙格-库塔方法（csRK）的注记。这些方法是对传统Runge-Kutta（RK）方法的一种有趣而又有创造性的扩展，为RK离散化提供了新的视角，并有可能扩大RK逼近理论在现代数学和工程领域的应用。研究csRK方法的一个突出优点是我们不需要研究与阶条件相关的多变量非线性代数方程的繁琐求解。本文将对近年来发展起来的csRK理论进行回顾、讨论和进一步推广。特别是，我们将把重点放在几何积分，包括辛方法，对称方法和能量保持的方法，在几何数值积分领域发挥了核心作用。
We provide a note on continuous-stage Runge–Kutta methods (csRK) for solving initial value problems of first-order ordinary differential equations. Such methods, as an interesting and creative extension of traditional Runge–Kutta (RK) methods, can give us a new perspective on RK discretization and it may enlarge the application of RK approximation theory in modern mathematics and engineering fields. A highlighted advantage of investigation of csRK methods is that we do not need to study the tedious solution of multi-variable nonlinear algebraic equations associated with order conditions. In this note, we will review, discuss and further promote the recently-developed csRK theory. In particular, we will place emphasis on geometric integrators including symplectic methods, symmetric methods and energy-preserving methods which play a central role in the field of geometric numerical integration.