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Optimal Runge-Kutta Stability Polynomials for Multidimensional High-Order Methods

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Abstract

In this paper we generate optimized Runge-Kutta stability polynomials for multidimensional discontinuous Galerkin methods recovered using the flux reconstruction approach. Results from linear stability analysis demonstrate that these stability polynomials can yield significantly larger time-step sizes for triangular, quadrilateral, hexahedral, prismatic, and tetrahedral elements with speedup factors of up to 1.97 relative to classical Runge-Kutta methods. Furthermore, performing optimization for multidimensional elements yields modest performance benefits for the triangular, prismatic, and tetrahedral elements. Results from linear advection demonstrate these schemes obtain their designed order of accuracy. Results from Direct Numerical Simulation (DNS) of a Taylor-Green vortex demonstrate the performance benefit of these schemes for unsteady turbulent flows, with negligible impact on accuracy.

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Data Statement

Data relating to the results in this manuscript can be downloaded from the publication’s website under a CC BY-NC-ND 4.0 license.

Change history

  • 02 October 2021

    The first and last name order of the author Siavash Hedayati Nasab was corrected.

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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], and the Fonds de Recherche Nature et Technologies (FRQNT) via the New University Researchers Start Up Program. 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.

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Authors and Affiliations

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

    Siavash Hedayati Nasab, Carlos A. Pereira & Brian C. Vermeire

Authors
  1. Siavash Hedayati Nasab
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  2. Carlos A. Pereira
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  3. Brian C. Vermeire
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Correspondence to Siavash Hedayati Nasab.

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Appendices

Appendices

Stability Plots

Fig. 11
figure 11

Plots of the \(\Delta t_{opt}\) as a function of the number of stages for quadrilateral elements with \(p =0\) and \(p=6\)

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Fig. 12
figure 12

Concave hull of semidiscrete eigenvalues scaled by \(\Delta t_{opt}\) (red) shown within the region of absolute stability of their corresponding stability polynomials for quadrilateral elements with \(p =4\)

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Fig. 13
figure 13

Plots of the \(\Delta t_{opt}\) as a function of the number of stages for triangular elements with \(p =0\) and \(p=6\)

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Fig. 14
figure 14

Concave hull of semidiscrete eigenvalues scaled by \(\Delta t_{opt}\)(red) shown within the region of absolute stability of their corresponding stability polynomials for triangular elements with \(p =4\)

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Fig. 15
figure 15

Plots of the \(\Delta t_{opt}\) as a function of the number of stages for hexahedral elements with \(p =0\) and \(p=6\)

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Fig. 16
figure 16

Concave hull of semidiscrete eigenvalues scaled by \(\Delta t_{opt}\)(red) shown within the region of absolute stability of their corresponding stability polynomials for hexahedral elements with \(p =4\)

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Fig. 17
figure 17

Plots of the \(\Delta t_{opt}\) as a function of the number of stages for prismatic elements with \(p =0\) and \(p=6\)

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Fig. 18
figure 18

Concave hull of semidiscrete eigenvalues scaled by \(\Delta t_{opt}\)(red) shown within the region of absolute stability of their corresponding stability polynomials for prismatic elements with \(p =4\)

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Fig. 19
figure 19

Plots of the \(\Delta t_{opt}\) as a function of the number of stages for tetrahedral elements with \(p =0\) and \(p=6\)

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Fig. 20
figure 20

Concave hull of semidiscrete eigenvalues scaled by \(\Delta t_{opt}\)(red) shown within the region of absolute stability of their corresponding stability polynomials for tetrahedral elements with \(p =4\)

Full size image

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Hedayati Nasab, S., Pereira, C.A. & Vermeire, B.C. Optimal Runge-Kutta Stability Polynomials for Multidimensional High-Order Methods. J Sci Comput 89, 11 (2021). https://doi.org/10.1007/s10915-021-01620-x

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