This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class of integrators are composition methods, in which one or several low-order schemes are composed to construct higher-order numerical approximations to the exact solution. We analyse in detail the order conditions that have to be satisfied by these classes of methods to achieve a given order, and provide some insight about their qualitative properties in connection with geometric numerical integration and the treatment of highly oscillatory problems. Since splitting methods have received considerable attention in the realm of partial differential equations, we also cover this subject in the present survey, with special attention to parabolic equations and their problems. An exhaustive list of methods of different orders is collected and tested on simple examples. Finally, some applications of splitting methods in different areas, ranging from celestial mechanics to statistics, are also provided.

中文翻译：

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微分方程的分裂方法


本概述主要介绍分裂方法，这是一类用于微分方程的数值积分器，可以将其细分为比原始系统更容易解决的不同问题。与此类积分器密切相关的是组合方法，其中组合一个或多个低阶方案以构造精确解的高阶数值近似。我们详细分析了此类方法为实现给定阶数而必须满足的阶数条件，并提供了有关它们与几何数值积分和高振荡问题的处理相关的定性特性的一些见解。由于分裂方法在偏微分方程领域受到了相当多的关注，我们在本次调查中也涵盖了这个主题，特别关注抛物线方程及其问题。收集了不同顺序的方法的详尽列表并在简单的示例上进行了测试。最后，还提供了分裂方法在从天体力学到统计学等不同领域的一些应用。