High-Order Accurate Implicit Runge-Kutta Schemes for the Simulation of Unsteady Flow Phenomena in Turbomachinery

Abstract

High-order accurate implicit Runge-Kutta schemes for the numerical integration of the compressible, unsteady Navier-Stokes equations in the context of a hybrid (structured and unstructured) flow solver are investigated and compared with more conventional implicit first- and second-order accurate time integration schemes. Three Explicit first stage, Stiffly accurate Diagonally Implicit Runge-Kutta (ESDIRK) schemes of second-, third- and fourth-order accuracy are considered. To solve the individual stages a direct (unfactored) approach in combination with the pseudo-time-stepping method is investigated. The effciency gains of the methods over lower order methods as well as the issues of robustness and complexity are discussed and analyzed in detail.