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

TL;DR

This paper introduces optimized Runge-Kutta methods with automatic step size control tailored for compressible CFD, enhancing efficiency and robustness across accuracy and stability regimes, especially in spectral element discretizations.

Contribution

It develops new error-control algorithms and controllers for Runge-Kutta methods, improving step size adaptation and performance in compressible CFD simulations.

Findings

Optimized methods outperform traditional CFL-based step size control.

Methods adaptively maintain high efficiency near maximum stable CFL numbers.

Numerical tests include complex industrial CFD applications.

Abstract

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 design new controllers for existing methods and for some new embedded Runge-Kutta pairs. We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice. 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 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.