  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.
