The use of operator-splitting methods to solve differential equations is widespread, but the methods are generally only defined for a given number of operators, most commonly two. Most operator-splitting methods are not generalizable to problems with $N$ operators for arbitrary $N$. In fact, there are only two known methods that can be applied to general $N$-split problems: the first-order Lie--Trotter (or Godunov) method and the second-order Strang (or Strang--Marchuk) method. In this paper, we derive two second-order operator-splitting methods that also generalize to $N$-split problems. These methods are complex valued but have positive real parts, giving them favorable stability properties, and require few sub-integrations per stage, making them computationally inexpensive. They can also be used as base methods from which to construct higher-order $N$-split operator-splitting methods with positive real parts. We verify the orders of accuracy of these new $N$-split methods and demonstrate their favorable efficiency properties against well-known real-valued operator-splitting methods on both real-valued and complex-valued differential equations.

翻译：算子分裂方法在求解微分方程中应用广泛，但这些方法通常仅针对特定数量的算子（最常见的是两个算子）定义。大多数算子分裂方法无法推广到具有任意$N$个算子的$N$分裂问题。事实上，目前仅知两种方法可应用于一般$N$分裂问题：一阶Lie--Trotter（或Godunov）方法和二阶Strang（或Strang--Marchuk）方法。本文推导了两种同样可推广至$N$分裂问题的二阶算子分裂方法。这些方法具有复数值但实部为正，因而具备良好的稳定性；同时每阶段仅需少量子积分步骤，计算成本较低。它们还可作为构造具有正实部的高阶$N$分裂算子分裂方法的基方法。我们验证了这些新型$N$分裂方法的精度阶数，并通过实值与复值微分方程的算例，证明了其相较于经典实值算子分裂方法更优的效能特性。