Optimized Runge-Kutta Methods with Automatic Step Size Control for\n Compressible Computational Fluid Dynamics

Abstract

We develop error-control based time integration algorithms for compressible\nfluid dynamics (CFD) applications and show that they are efficient and robust\nin both the accuracy-limited and stability-limited regime. Focusing on\ndiscontinuous spectral element semidiscretizations, we design new controllers\nfor existing methods and for some new embedded Runge-Kutta pairs. We\ndemonstrate the importance of choosing adequate controller parameters and\nprovide a means to obtain these in practice. We compare a wide range of\nerror-control-based methods, along with the common approach in which step size\ncontrol is based on the Courant-Friedrichs-Lewy (CFL) number. The optimized\nmethods give improved performance and naturally adopt a step size close to the\nmaximum stable CFL number at loose tolerances, while additionally providing\ncontrol of the temporal error at tighter tolerances. The numerical examples\ninclude challenging industrial CFD applications.\n