Towards New High-Order Operator Splitting Time-Integration Methods

Abstract

Operator splitting (OS) methods represent a powerful strategy to solve an extensive range of mathematical models in the form of differential equations. They have a long and celebrated history, having been successfully used for well over half a century to provide efficient numerical solutions to challenging problems. In fact, OS methods are often the only viable way to solve many problems in practice.

The simplest, and perhaps, most well-known OS methods are Lie--Trotter--Godunov and the Strang--Marchuk methods. They compute a numerical solution that is first-, and second-order accurate in time, respectively. OS methods can be derived by imposing order conditions using the Campbell--Baker--Hausdorff formula. It follows that, by setting the appropriate order conditions, it is possible to derive OS methods of any desired order. An important observation regarding OS methods with order higher than two is that, according to the Sheng--Suzuki theorem, at least one of their defining coefficients must be negative. Therefore, the time integration with OS methods of order higher than two has not been considered suitable to solve deterministic parabolic problems, because the necessary backward time integration would cause instabilities.

Throughout this thesis, we focus our attention on high-order (i.e., order higher than two) OS methods. We successfully assess the convergence properties of some higher-order OS methods on diffusion-reaction problems describing cardiac electrophysiology and on an advection-diffusion-reaction problem describing chemical combustion. Furthermore, we compare the efficiency performance of higher-order methods to second-order methods. For all the cases considered, we confirm an improved efficiency performance compared to methods of lower order.

Next, we observe how, when using OS and Runge--Kutta type methods to advance the time integration, we can construct a unique extended Butcher tableau with a similar structure to the ones describing Generalized Additive Runge--Kutta (GARK) methods. We define a combination of methods to be OS-GARK methods. We apply linear stability analysis to OS-GARK methods; this allows us to conveniently analyze the stability properties of any combination of OS and Runge--Kutta methods. Doing so, we are able to perform an eigenvalue analysis to understand and improve numerically unstable solutions.