Operator splitting is a popular divide-and-conquer strategy for solving differential equations. Typically, the right-hand side of the differential equation is split into a number of parts that are then integrated separately. Many methods are known that split the right-hand side into two parts. This approach is limiting, however, and there are situations when 3-splitting is more natural and ultimately more advantageous. The second-order Strang operator-splitting method readily generalizes to a right-hand side splitting into any number of operators. It is arguably the most popular method for 3-splitting because of its efficiency, ease of implementation, and intuitive nature. Other 3-splitting methods exist, but they are less well-known, and \rev{analysis and} evaluation of their performance in practice are scarce. We demonstrate the effectiveness of some alternative 3-split, second-order methods to Strang splitting on two problems: the reaction-diffusion Brusselator, which can be split into three parts that each have closed-form solutions, and the kinetic Vlasov--Poisson equations that is used in semi-Lagrangian plasma simulations. We find alternative second-order 3-operator-splitting methods that realize efficiency gains of 10\%--20\% over traditional Strang splitting. Our analysis for the practical assessment of efficiency of operator-splitting methods includes the computational cost of the integrators and can be used in method design.

翻译：算子分裂是求解微分方程的一种流行的分治策略。通常，微分方程的右端项被分裂为多个部分，随后对各部分分别进行积分。已知有许多方法可将右端项分裂为两部分，然而这种处理方式存在局限性，而在某些情况下三分裂更为自然且最终更具优势。二阶Strang算子分裂方法可自然地推广至任意数量算子的右端项分裂。由于其高效性、易于实现及直观的特性，这可以说是最流行的三分裂方法。其他三分裂方法也存在，但知名度较低，且对其实际性能的分析与评估十分匮乏。我们在两个问题上论证了若干替代Strang分裂的（三算子）二阶方法的有效性：反应扩散型Brusselator模型（其可分裂为三个均具有闭式解的部分），以及用于半拉格朗日等离子体模拟的动力学Vlasov-Poisson方程。我们发现，替代性二阶三算子分裂方法相比传统Strang分裂可实现10%-20%的效率提升。我们对算子分裂方法效率的实践评估分析包含了积分器的计算成本，可用于方法设计。