I gave a talk Combining Analysis and Data: Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics in the Applied Mathematics Seminar, University of Münster, on Wednesday, 2021-07-07, 14:00 CEST. The official announcement is available online.
We develop error-control based time integration algorithms for compressible fluid dynamics (CFD) applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime, focusing on discontinuous spectral element semidiscretizations. We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice by combining analysis with a data-driven approach, which we apply to design new controllers for existing methods and for some new embedded Runge-Kutta pairs. We compare a wide range of error-control-based methods, along with the common approach in which step size control is based on the Courant-Friedrichs-Lewy (CFL) number. The new optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances, while additionally providing control of the temporal error at tighter tolerances. The numerical examples include challenging industrial CFD applications.
A preprint of this joint work with Lisandro Dalcin, Matteo Parsani, and David Ketcheson is available on arXiv.org.