Vermeire, B.C., Loppi, N.A. and Vincent, P.E. (2019) Optimal Runge-Kutta Schemes for Pseudo Time-Stepping with High-Order Unstructured Methods. Journal of Computational Physics . ISSN 00219991 (In Press)
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Official URL: http://dx.doi.org/10.1016/j.jcp.2019.01.003
Abstract
In this study we generate optimal Runge-Kutta (RK) schemes for converging the Artificial Compressibility Method (ACM) using dual time-stepping with high-order unstructured spatial discretizations. We present optimal RK schemes with between s = 2 and s = 7 stages for Spectral Difference (SD) and Discontinuous Galerkin (DG) discretizations obtained using the Flux Reconstruction (FR) approach with solution polynomial degrees of k = 1 to k = 8. These schemes are optimal in the context of linear advection with predicted speedup factors in excess of 1.80× relative to a classical RK4,4 scheme. Speedup factors of between 1.89× and 2.11× are then observed for incompressible Implicit Large Eddy Simulation (ILES) of turbulent flow over an SD7003 airfoil. Finally, we demonstrate the utility of the schemes for incompressible ILES of a turbulent jet, achieving good agreement with experimental data. The results demonstrate that the optimized RK schemes are suitable for simulating turbulent flows and can achieve significant speedup factors when converging the ACM using dual time-stepping with high-order unstructured spatial discretizations.
| Divisions: | Concordia University > Gina Cody School of Engineering and Computer Science > Mechanical, Industrial and Aerospace Engineering |
|---|---|
| Item Type: | Article |
| Refereed: | Yes |
| Authors: | Vermeire, B.C. and Loppi, N.A. and Vincent, P.E. |
| Journal or Publication: | Journal of Computational Physics |
| Date: | 30 January 2019 |
| Funders: |
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| Digital Object Identifier (DOI): | 10.1016/j.jcp.2019.01.003 |
| ID Code: | 984983 |
| Deposited By: | Michael Biron |
| Deposited On: | 08 Feb 2019 22:28 |
| Last Modified: | 30 Jan 2021 02:00 |
References:
A.J. Chorin. A numerical methods for solving incompressible viscous flow problems. Journal of Computational Physics, 135:118 – 125, 1997.A. Jameson. Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. In 10th AIAA Computational Fluid Dynamics Conference, Honolulu, HI, 1991. American Institute of Aeronautics and Astronautics. AIAA Paper 1991-1596.
A. Jameson and S. Yoon. Lower-upper implicit schemes with multiple grids for the Euler equations. AIAA Journal, 25(7):929 – 935, 1987.
Christopher Cox, Chunlei Liang, and Michael W. Plesniak. A high-order solver for unsteady incompressible navier–stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping. Journal of Computational Physics, 314:414 – 435, 2016.
B. Cockburn and C. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. general framework. Mathematics of Computation, 52(186):411–435, 1989.
B. Cockburn, S. Hou, and C. Shu. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. the multidimensional case. Mathematics of Computation, 54(190):545–581, 1990.
Y. Liu, M. Vinokur, and Z. J. Wang. Discontinuous spectral difference method for conservation laws on unstructured grids. In C. Groth and D. W. Zingg, editors, Computational Fluid Dynamics 2004, pages 449–454. Springer Berlin Heidelberg, January 2006.
D. A. Kopriva. A conservative staggered-grid Chebyshev multidomain method for compressible flows. II: a semi-structured method. Technical report, March 1996.
H. T. Huynh. A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. In 18th AIAA Computational Fluid Dynamics Conference, Miami, FL, June 2007. American Institute of Aeronautics and Astronautics. AIAA Paper 2007-4079.
B. C. Vermeire, J. S. Cagnone, and S. Nadarajah. ILES using the correction procedure via reconstruction scheme. In 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Grapevine, TX, January 2013. American Institute of Aeronautics and Astronautics. AIAA Paper 2013-1001.
B. C. Vermeire, S. Nadarajah, and P. G. Tucker. Canonical test cases for high-order unstructured implicit large eddy simulation. In AIAA Science and Technology Forum and Exposition (SciTech 2014), National Harbor, Maryland, January 2014. American Institute of Aeronautics and Astronautics. AIAA Paper 2014-0935.
B. C. Vermeire, S. Nadarajah, and P. G. Tucker. Implicit large eddy simulation using the high-order correction procedure via reconstruction scheme. International Journal for Numerical Methods in Fluids, 82(5):231–260, 2016. fld.4214.
B.C. Vermeire, F.D. Witherden, and P.E. Vincent. On the utility of GPU accelerated high-order methods for unsteady flow simulations: A comparison with industry-standard tools. Journal of Computational Physics, 334:497 – 521, 2017.
F. Bassi, A. Crivellini, D.A. Di Pietro, and S. Rebay. An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible navier–stokes equations. Journal of Computational Physics, 218(2):794 – 815, 2006.
Khosro Shahbazi, Paul F. Fischer, and C. Ross Ethier. A high-order discontinuous Galerkin method for the unsteady incompressible navier–stokes equations. Journal of Computational Physics, 222(1):391 – 407, 2007.
B. Cockburn N. Nguyen, J. Peraire. A hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations. In 48th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, Orlando, FL, 2010. American Institute of Aeronautics and Astronautics. AIAA Paper 2010-362.
P. Vincent, F. Witherden, B. Vermeire, J.S. Park, and A. Iyer. Towards green aviation with python at petascale. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC '16, pages 1:1–1:11, Piscataway, NJ, USA, 2016. IEEE Press.
N.A. Loppi, F.D. Witherden, A. Jameson, and P.E. Vincent. A high-order cross-platform incompressible Navier–Stokes solver via artificial compressibility with application to a turbulent jet. Computer Physics Communications, 233:193–205, 2018.
E. Hairer and H. Wanner. Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic Problems. Springer, 2 edition, 1996.
D.I. Ketcheson and A.J. Ahmidia. Optimal stability polynomials for numerical integration of initial value problems. Communications in Applied Mathematics and Computational Science, 7(2):247 – 271, 2012.
D.I. Ketcheson, M. Parsani, and A.J. Ahmadia. Rk-opt: Software for the design of Runge-Kutta methods, Apr 2013.
M. Parsani, D.I. Ketcheson, and W. Deconinck. Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM Journal on Scientific Computing, 35(2):957 – 986, 2013.
E.J. Kubatko, B.A. Yeager, and D.I. Ketcheson. Optimal strong-stability-preserving runge–kutta time discretizations for discontinuous Galerkin methods. Journal of Scientific Computing, 60(2):313–344, Aug 2014.
Matlab 2016a, 2016. The MathWorks, Natick, MA, USA.
M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx, March 2014.
M. Grant and S. Boyd. Graph implementations for nonsmooth convex programs. In V. Blondel, S. Boyd, and H. Kimura, editors, Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, pages 95–110. Springer-Verlag Limited, 2008. http://stanford.edu/~boyd/graph_dcp.html.
P. E. Vincent, P. Castonguay, and A. Jameson. Insights from von Neumann analysis of high-order flux reconstruction schemes. Journal of Computational Physics, 230(22):8134–8154, September 2011.
B.C. Vermeire and P.E. Vincent. On the behaviour of fully-discrete flux reconstruction schemes. Computer Methods in Applied Mechanics and Engineering, 315:1053 – 1079, 2017.
B.C. Vermeire and P.E. Vincent. On the properties of energy stable flux reconstruction schemes for implicit large eddy simulation. Journal of Computational Physics, 327:368 – 388, 2016.
P. E. Vincent, P. Castonguay, and A. Jameson. A new class of high-order energy stable flux reconstruction schemes. Journal of Scientific Computing, 47(1):50–72, April 2011.
M. S. Selig, J. F. Donovan, and D. B. Fraser. Airfoils at Low Speed. Stokely, 1989.
F. D. Witherden, A. M. Farrington, and P. E. Vincent. PyFR: An Open Source Framework for Solving Advection-Diffusion Type Problems on Streaming Architectures using the Flux Reconstruction Approach. Computer Physics Communications, 185(11):3028–3040, November 2014.
M. R. Visbal, R. E. Gordnier, and M. C. Galbraith. High-fidelity simulations of moving and flexible airfoils at low reynolds numbers. Experiments in Fluids, 46(5):903–922, May 2009.
D. J. Garmann, M. R. Visbal, and P. D. Orkwis. Comparative study of implicit and subgrid-scale model large-eddy simulation techniques for low-reynolds number airfoil applications. International Journal for Numerical Methods in Fluids, 71(12):1546–1565, 2013.
A.D. Beck, T. Bolemann, D. Flad, H. Frank, G.J. Gassner, F. Hindenlang, and C.D. Munz. High-order discontinuous Galerkin spectral element methods for transitional and turbulent flow simulations. International Journal for Numerical Methods in Fluids, 76(8):522–548, 2014.
G. Lipari and P. K. Stansby. Review of experimental data on incompressible turbulent round jets. Flow, Turbulence and Combustion, 87(1):79–114, 2011.
B. J. Boersma. Numerical simulation of the noise generated by a low Mach number, low Reynolds number jet. Fluid Dynamics Research, 35(6):425–447, 2004.
C. Bogey and C. Bailly. Turbulence and energy budget in a self-preserving round jet: direct evaluation using large eddy simulation. Journal of Fluid Mechanics, 627:129, 2009.
N. R. Panchapakesan and J. L. Lumley. Turbulence measurements in axisymmetric jets of air and helium. I - Air jet. II - Helium jet. Journal of Fluid Mechanics, 246(1993):197 – 223, 1993.
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