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：本综述聚焦于分裂方法——一类专为可分解为若干较原始系统更易求解的子问题的微分方程设计的数值积分器。与此类积分器密切相关的是组合方法，即通过组合一个或多个低阶格式构建高阶数值逼近精确解。我们详细分析了这些方法类达到给定阶次所需满足的阶条件，并深入探讨其与几何数值积分及高频振荡问题处理相关的定性性质。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，重点关注抛物型方程及其相关问题。文中系统收集了不同阶次的各类方法，并通过简单算例进行验证。最后，还展示了分裂方法从天体力学到统计学等不同领域的应用实例。