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：本文综述致力于分裂方法——一类数值积分器，专门用于可分解成多个比原系统更易求解的子问题的微分方程。与此类积分器紧密相关的是组合方法，通过组合一个或多个低阶格式构造高阶数值逼近精确解。我们详细分析了这些方法达到指定阶次必须满足的阶条件，并深入探讨了其在几何数值积分与高频振荡问题处理中的定性性质。鉴于分裂方法在偏微分方程领域已获得广泛关注，本综述亦涵盖该主题，特别关注抛物线方程及其相关问题。我们收集了不同阶次的完备方法列表，并通过简单算例进行验证。最后，还介绍了分裂方法从天体力学到统计学等不同领域的应用实例。