High-order implicit time integration for unsteady compressible fluid flow simulation

This paper presents an overview of high-order implicit time integration methods and their associated properties with a specific focus on their application to computational fluid dynamics. A framework is constructed for the development and optimization of general implicit time integration methods, specifically including linear multistep, Runge-Kutta, and multistep Runge-Kutta methods. The analysis and optimization capabilities of the framework are verified by rederiving methods with known coeffcients. The framework is then applied to the derivation of novel singly-diagonally-implicit Runge-Kutta methods, explicit-first-stage singly-diagonally implicit Runge-Kutta methods, and singly-diagonally-implicit multistep Runge-Kutta methods. The fourth-order methods developed have similar effciency to contemporary methods; however a fifth-order explicit-first-stage singly-diagonally-implicit Runge-Kutta method is obtained with higher relative effciency. This is confirmed with simulations of van der Pol's equation.