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 analyze 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.

翻译：本综述专注于分裂方法——这类数值积分器专为可将原系统分解为更易求解子问题的微分方程而设计。与此类积分器密切相关的还有组合方法，其通过组合一个或多个低阶格式来构造高阶数值逼近精确解。我们详细分析了这类方法达到指定阶数所必须满足的阶条件，并深入探讨了其在几何数值积分与高频振荡问题处理方面的定性特性。鉴于分裂方法在偏微分方程领域已获得广泛关注，本次综述亦涵盖该主题，特别关注抛物型方程及其相关问题。我们整理并测试了不同阶数的详尽可能的方法列表，并通过简单算例进行了验证。最后，本文还展示了分裂方法在天体力学与统计学等不同领域中的应用实例。