https://hdl.handle.net/10388/16266

Operator-splitting methods are a popular divide-and-conquer approach for the numerical solution of differential equations. They split differential equations into simpler subsystems, discretize each subsystem separately, and combine the discretizations to obtain a solution to the original problem. Operator-splitting methods allow us to take advantage of the characteristics of each subsystem, such as stiffness or the existence of exact solutions. Operator-splitting methods are prevalent in multi-physic applications because of various difficulties when trying to solve the system monolithically. Such applications also require numerical methods to preserve qualitative properties such as positivity or the conservation of total energy. In this thesis, we study the effect of operator-splitting methods on qualitative property preservation in the following two aspects: backward-in-time integration of high-order operator methods and different splitting strategies. We propose guidelines for operator-splitting schemes to satisfy desired qualitative properties. Although many applications use low-order methods and focus on splitting the system into two or three subsystems, the complexity of some applications inspires exploring new directions of operator-splitting. Splitting methods for general N-split systems are limited due to the complexity deriving them from order conditions. A general analysis for N-split methods is lacking. In this thesis, we investigate order conditions and stability of N-split methods and propose some novel complex-valued N-split methods with desirable features. Moreover, we examine the performance of complex-coefficient methods compared to real-coefficient methods. Moreover, despite the availability of high-order methods, many applications resort to lower-order methods due to concerns about stability arising from backward-in-time integration. Because of the lack of general implementation of an operator-splitting library, most applications use only the first-order Godunov method or the second-order Strang--Marchuk method. However, operator-splitting methods can be tailored around specific differential equations to achieve better efficiency and stability. In this thesis, we aim to propose and test guidelines for designing operator-splitting methods and the use of operator-splitting methods to optimize efficiency. Additionally, to facilitate the practical implementation of these methods, we introduce the software, providing a framework for general experimentation with splitting methods.

https://hdl.handle.net/10388/16266