## Abstract

In compressible computational fluid dynamics, the step size of explicit time integration schemes is often constrained by stability when high-order accurate spatial discretizations are used. We report a set of new optimized explicit Runge–Kutta schemes for the integration of systems of ordinary differential equations arising from the spatial discretization of wave propagation problems with high-order entropy stable collocated discontinuous Galerkin methods. The eigenvalues of the discrete spatial operator for the advection equation and the propagation of an isentropic vortex with the compressible Euler equations for various values of the problems' parameters are used to optimize the stability region of the proposed time integration schemes. To demonstrate the efficiency and the robustness of the methods, we solve the compressible turbulent flow past the Valeo controlled-diffusion airfoil and a delta wing at a Reynolds number of 8.3×10^{5} and 10^{6}, respectively. A thorough analysis of the performance of the two families of optimized schemes revealed that methods generated using the spectra of the vortex problem are 6-to-20% faster than methods constructed using the spectra of the advection equation. Compared to widely used explicit Runge–Kutta schemes, the methods designed using the spectra of the vortex problem yield a time-to-solution saving of approximately 6-to-38%. For large-scale time-dependent partial differential equations computations, these gains mean saving hundreds of thousands if not millions of core hours. In addition, the new methods can be effectively and efficiently applied to integrate systems of ordinary differential equations arising from a wide range of spatial discretization, including discontinuous Galerkin spectral element methods, spectral difference methods, and flux reconstruction methods.

Original language | English (US) |
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Pages (from-to) | 1-17 |

Number of pages | 17 |

Journal | Computers and Mathematics with Applications |

Volume | 118 |

DOIs | |

State | Published - Jul 15 2022 |

### Bibliographical note

Funding Information:The work described in this paper was supported by King Abdullah University of Science and Technology through the award OSR-2019-CCF-3666 . The authors are also thankful for the computing resources of the Supercomputing Laboratory and the Extreme Computing Research Center at King Abdullah University of Science and Technology.

Publisher Copyright:

© 2022 The Author(s)

## Keywords

- Advection equation
- Collocated discontinuous Galerkin methods
- Compressible Euler and Navier–Stokes equations
- Optimized explicit Runge–Kutta schemes
- Unstructured grids

## ASJC Scopus subject areas

- Modeling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics