一对二阶复值 N 分裂算子分裂方法
A pair of Second-order complex-valued, N-split operator-splitting methods
AI总结：
针对一般 N 个算子的分裂问题，论文提出两种二阶复值方法，兼具正实部稳定性与低计算成本，并可用于构造更高阶方法。
摘要：
使用算子分裂方法求解微分方程十分普遍，但这些方法通常只针对给定数量的算子定义，最常见的是两个。大多数算子分裂方法不能推广到具有任意 N 个算子的问题。事实上，已知可用于一般 N 分裂问题的方法只有两种：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到 N 分裂问题的二阶算子分裂方法。这些方法是复值的，但具有正实部，因此具有良好的稳定性；并且每一级只需少量子积分，计算成本较低。它们还可用作基础方法，以构造具有正实部的更高阶 N 分裂算子分裂方法。我们验证了这些新的 N 分裂方法的精度阶数，并在实值和复值微分方程上，相对于知名的实值算子分裂方法展示了其良好的效率特性。
英文摘要：
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.
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