Assessment of efficiency, robustness and accuracy effects of the discrete per-stage formulation of implicit time integration when solving inexactly

Abstract

This thesis focuses on two main objectives to improve unsteady computational fluid dynamics (CFD) simulations. The first was to implement and test an alternative formulation for combining iterative stage state solutions in Diagonally Implicit Runge-Kutta (DIRK) methods. The new formulation demonstrated comparable performance to the default method at high accuracy. With larger time step sizes, it provided slight improvements in computational cost. It also managed to converge with fewer iterations, unlike the default approach. However, additional errors were observed when reducing time step sizes. The second objective was to implement and evaluate an adaptive time-stepping algorithm using embedded Runge-Kutta methods to estimate errors and adjust time step sizes respectively. This method reduced simulation runtime by 31.65 % for the case of a gust encounter compared to a constant time-stepping approach, while maintaining similar accuracy. It has also been shown that the algorithm can be more accurate at the same computational cost and can be used with test cases having a more constant computational demand, compared to a gust encounter.