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Fully-Discrete Analysis of High-Order Spatial Discretizations with Optimal Explicit Runge–Kutta Methods

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Abstract

High-order unstructured methods have become a popular choice for the simulation of complex unsteady flows. Flux reconstruction (FR) is a high-order spatial discretization method, which has been found to be particularly accurate for scale-resolving simulations of complex phenomena. In addition, it has been shown to provide sufficient dissipation for implicit large-eddy simulation (ILES). In conjunction with an FR discretization, an appropriate temporal scheme must be chosen. A common choice is explicit schemes due to their efficiency and ease of implementation. However, these methods usually require a small time-step size to remain stable. Recently, the development of optimal explicit Runge–Kutta (OERK) schemes has enabled stable simulations with larger time-step sizes. Hence, we analyze the fully-discrete properties of the FR method with OERK temporal schemes. We show results for first, second, third, fourth and eighth-order OERK schemes. We observe that OERK schemes modify the spectral behaviour of the semidiscretization. In particular, dissipation decreases in the region of high wavenumbers. We observe that higher-order OERK schemes require a smaller time step than the low-order schemes. However, they follow the dispersion relations of the FR scheme for a larger range of wavenumbers. We validate our analysis with simple advection test cases. It was observed that first and second-degree temporal schemes introduce a relatively large amount of error in the solutions. A one-dimensional ILES test case showed that, as long as the time-step size is not in the vicinity of the stability limit, results are generally similar to classical RK schemes.

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Acknowledgements

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [RGPAS-2017-507988, RGPIN-2017-06773]. This research was enabled in part by support provided by Calcul Quebec (www.calculquebec.ca), WestGrid (www.westgrid.ca), SciNet (www.scinethpc.ca), and Compute Canada (www.computecanada.ca) via a Resources for Research Groups allocation. We would also like to thank the anonymous reviewers whose comments improved the quality of this manuscript.

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  1. Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal, QC, Canada

    Carlos A. Pereira & Brian C. Vermeire

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  1. Carlos A. Pereira
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Correspondence to Carlos A. Pereira.

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Spectral Properties of Fully Discrete FR-\(P_{s,4}\) Schemes

Spectral Properties of Fully Discrete FR-\(P_{s,4}\) Schemes

See Figs. 19, 20, 21 and 22.

Fig. 19
figure 19

Spectral properties of \(P_{5,4}\) schemes at different CFL numbers \(\tau \). Properties of the semidiscretization are shown in a dashed line

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Fig. 20
figure 20

Spectral properties of \(P_{6,4}\) schemes at different CFL numbers \(\tau \). Properties of the semidiscretization are shown in a dashed line

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Fig. 21
figure 21

Spectral properties of \(P_{7,4}\) schemes at different CFL numbers \(\tau \). Properties of the semidiscretization are shown in a dashed line

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Fig. 22
figure 22

Spectral properties of \(P_{8,4}\) schemes at different CFL numbers \(\tau \). Properties of the semidiscretization are shown in a dashed line

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Pereira, C.A., Vermeire, B.C. Fully-Discrete Analysis of High-Order Spatial Discretizations with Optimal Explicit Runge–Kutta Methods. J Sci Comput 83, 63 (2020). https://doi.org/10.1007/s10915-020-01243-8

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