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Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation

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Abstract

We conduct a comparative study of the Jacobian-free linearly implicit Rosenbrock–Wanner (ROW) methods, the explicit first stage, singly diagonally implicit Runge–Kutta (ESDIRK) methods, and the second-order backward differentiation formula (BDF2) for the high-order flux reconstruction/correction procedure via reconstruction solution of the unsteady Navier–Stokes equations. Pseudo-transient continuation is employed to solve the nonlinear equation at each stage of ESDIRK (excluding the first stage) and each step of BDF2. A Jacobian-free implementation of the restarted generalized minimal residual method solver is employed with a low storage element-Jacobi preconditioner to solve the linear system at each stage of ROW and each pseudo time iteration of ESDIRK and BDF2. Several numerical experiments, including both laminar and turbulent flow simulations, are conducted to carry out the comparison. We observe that the multistage ROW2 and ESDIRK2 are more efficient than the multistep BDF2, and higher-order implicit time integrators are more efficient than lower-order ones. In general, the ESDIRK method allows a larger physical time step size for unsteady flow simulation than the ROW method when the element-Jacobi preconditioner is employed, especially for wall-bounded flows; and the ROW method can be more efficient than the ESDIRK method when the time step size is refined.

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Acknowledgements

The authors gratefully acknowledge the support of the Office of Naval Research through the award N00014-16-1-2735, and the faculty startup support from the department of mechanical engineering at the University of Maryland, Baltimore County (UMBC). The hardware used in the computational studies is part of the UMBC High Performance Computing Facility (HPCF). The facility is supported by the U.S. National Science Foundation through the MRI program (Grant Nos. CNS-0821258, CNS-1228778, and OAC-1726023) and the SCREMS program (Grant No. DMS-0821311), with additional substantial support from UMBC.

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  1. Department of Mechanical Engineering, University of Maryland, Baltimore County, Baltimore, MD, 21250, USA

    Lai Wang & Meilin Yu

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  1. Lai Wang
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Appendix

Appendix

The coefficients of ESDIRK and ROW methods investigated in this study are documented here for completeness.

See Tables 5 and 6.

Table 5 Coefficients of ESDIRK methods
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Table 6 Coefficients of ROW methods
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Wang, L., Yu, M. Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation. J Sci Comput 83, 39 (2020). https://doi.org/10.1007/s10915-020-01222-z

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