A pair of Second-order complex-valued, N-split operator-splitting methods

TL;DR

This paper introduces two novel second-order complex-valued operator-splitting methods that generalize to N-operator problems, offering improved stability and efficiency for solving differential equations.

Contribution

The paper develops two second-order N-split operator-splitting methods that are complex-valued with positive real parts, extending the applicability of operator-splitting techniques to arbitrary N.

Findings

Methods are verified to be second-order accurate.

New methods demonstrate superior efficiency over existing real-valued methods.

Applicable to both real-valued and complex-valued differential equations.

Abstract

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.