A second-order ensemble method based on a blended backward differentiation formula timestepping scheme for time-dependent Navier-Stokes equations

Jiang, Nan

doi:10.1002/num.22070

Title: A second-order ensemble method based on a blended backward differentiation formula timestepping scheme for time-dependent Navier-Stokes equations

Journal Article · Tue May 31 00:00:00 EDT 2016 · Numerical Methods for Partial Differential Equations

DOI:https://doi.org/10.1002/num.22070· OSTI ID:1533198

Jiang, Nan ^[1]

Florida State Univ., Tallahassee, FL (United States)

Here, we present a second-order ensemble method based on a blended three-step backward differentiation formula (BDF) timestepping scheme to compute an ensemble of Navier–Stokes equations. Compared with the only existing second-order ensemble method that combines the two-step BDF timestepping scheme and a special explicit second-order Adams–Bashforth treatment of the advection term, this method is more accurate with nominal increase in computational cost. We give comprehensive stability and error analysis for the method. Numerical examples are also provided to verify theoretical results and demonstrate the improved accuracy of the method.

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Research Organization:: Florida State Univ., Tallahassee, FL (United States)

Sponsoring Organization:: USDOE Office of Science (SC); US Air Force Office of Scientific Research (AFORSR)

Grant/Contract Number:: SC0010678; FA9550-15-1-0001

OSTI ID:: 1533198

Journal Information:: Numerical Methods for Partial Differential Equations, Vol. 33, Issue 1; ISSN 0749-159X

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 35 works

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References (26)

Stabilized finite element schemes with LBB-stable elements for incompressible flows Gelhard, Tobias; Lube, Gert; Olshanskii, Maxim A. Journal of Computational and Applied Mathematics, Vol. 177, Issue 2 https://doi.org/10.1016/j.cam.2004.09.017	journal	May 2005
The Mathematical Theory of Finite Element Methods Brenner, Susanne C.; Scott, L. Ridgway Texts in Applied Mathematics https://doi.org/10.1007/978-0-387-75934-0	book	January 2008
Analysis of Model Variance for Ensemble Based Turbulence Modeling Jiang, Nan; Kaya, Songul; Layton, William Computational Methods in Applied Mathematics, Vol. 15, Issue 2 https://doi.org/10.1515/cmam-2014-0029	journal	January 2015
On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations Baker, Garth A.; Dougalis, Vassilios A.; Karakashian, Ohannes A. Mathematics of Computation, Vol. 39, Issue 160 https://doi.org/10.1090/S0025-5718-1982-0669634-0	journal	January 1982
Re-evaluation of an Optimized Second Order Backward Difference (BDF2OPT) Scheme for Unsteady Flow Applications Vatsa, Veer; Carpenter, Mark; Lockard, David 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition https://doi.org/10.2514/6.2010-122	conference	June 2010
Time relaxation algorithm for flow ensembles: Time Relaxation Algorithm Takhirov, Aziz; Neda, Monika; Waters, Jiajia Numerical Methods for Partial Differential Equations, Vol. 32, Issue 3 https://doi.org/10.1002/num.22024	journal	October 2015
An Algorithm for fast Calculation of flow Ensembles Jiang, Nan; Layton, William International Journal for Uncertainty Quantification, Vol. 4, Issue 4 https://doi.org/10.1615/Int.J.UncertaintyQuantification.2014007691	journal	January 2014
The Singular-Vector Structure of the Atmospheric Global Circulation Buizza, R.; Palmer, T. N. Journal of the Atmospheric Sciences, Vol. 52, Issue 9 https://doi.org/10.1175/1520-0469(1995)052<1434:TSVSOT>2.0.CO;2	journal	May 1995
Convergence properties of block GMRES and matrix polynomials Simoncini, V.; Gallopoulos, E. Linear Algebra and its Applications, Vol. 247 https://doi.org/10.1016/0024-3795(95)00093-3	journal	November 1996
Numerical analysis of two ensemble eddy viscosity numerical regularizations of fluid motion: Two Ensemble Eddy Viscosity Numerical Regularizations Jiang, Nan; Layton, William Numerical Methods for Partial Differential Equations, Vol. 31, Issue 3 https://doi.org/10.1002/num.21908	journal	July 2014
Integrated Design of an Active Flow Control System Using a Time-Dependent Adjoint Method Nielsen, E. J.; Jones, W. T. Mathematical Modelling of Natural Phenomena, Vol. 6, Issue 3 https://doi.org/10.1051/mmnp/20116306	journal	January 2011
Exact fully 3D Navier-Stokes solutions for benchmarking Ethier, C. Ross; Steinman, D. A. International Journal for Numerical Methods in Fluids, Vol. 19, Issue 5 https://doi.org/10.1002/fld.1650190502	journal	September 1994
Velocity–vorticity–helicity formulation and a solver for the Navier–Stokes equations Olshanskii, Maxim A.; Rebholz, Leo G. Journal of Computational Physics, Vol. 229, Issue 11 https://doi.org/10.1016/j.jcp.2010.02.012	journal	June 2010
New development in freefem++ Hecht, F. Journal of Numerical Mathematics, Vol. 20, Issue 3-4 https://doi.org/10.1515/jnum-2012-0013	journal	January 2012
Partially Implicit BDF2 Blends for Convection Dominated Flows Hundsdorfer, Willem SIAM Journal on Numerical Analysis, Vol. 38, Issue 6 https://doi.org/10.1137/S0036142999364741	journal	January 2001
A Connection Between Scott–Vogelius and Grad-Div Stabilized Taylor–Hood FE Approximations of the Navier–Stokes Equations Case, Michael A.; Ervin, Vincent J.; Linke, Alexander SIAM Journal on Numerical Analysis, Vol. 49, Issue 4 https://doi.org/10.1137/100794250	journal	January 2011
Predicting probability distributions for surf height using an ensemble of mixture density networks Carney, Michael; Cunningham, Pádraig; Dowling, Jim Proceedings of the 22nd international conference on Machine learning - ICML '05 https://doi.org/10.1145/1102351.1102366	conference	January 2005
A block conjugate gradient method applied to linear systems with multiple right-hand sides Feng, Y. T.; Owen, D. R. J.; Perić, D. Computer Methods in Applied Mechanics and Engineering, Vol. 127, Issue 1-4 https://doi.org/10.1016/0045-7825(95)00832-2	journal	November 1995
A Higher Order Ensemble Simulation Algorithm for Fluid Flows Jiang, Nan Journal of Scientific Computing, Vol. 64, Issue 1 https://doi.org/10.1007/s10915-014-9932-z	journal	October 2014
Error of the two-step BDF for the incompressible Navier-Stokes problem Emmrich, Etienne ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 38, Issue 5 https://doi.org/10.1051/m2an:2004037	journal	September 2004
A low order Galerkin finite element method for the Navier–Stokes equations of steady incompressible flow: a stabilization issue and iterative methods Olshanskii, Maxim A. Computer Methods in Applied Mechanics and Engineering, Vol. 191, Issue 47-48 https://doi.org/10.1016/S0045-7825(02)00513-3	journal	November 2002
A block QMR algorithm for non-Hermitian linear systems with multiple right-hand sides Freund, Roland W.; Malhotra, Manish Linear Algebra and its Applications, Vol. 254, Issue 1-3 https://doi.org/10.1016/S0024-3795(96)00529-0	journal	March 1997
On stability and convergence of projection methods based on pressure Poisson equation Guermond, J. -L.; Quartapelle, L. International Journal for Numerical Methods in Fluids, Vol. 26, Issue 9 https://doi.org/10.1002/(SICI)1097-0363(19980515)26:9<1039::AID-FLD675>3.0.CO;2-U	journal	May 1998
Sensitivity Analysis of Convection of the 24 May 2002 IHOP Case Using Very Large Ensembles Martin, William J.; Xue, Ming Monthly Weather Review, Vol. 134, Issue 1 https://doi.org/10.1175/MWR3061.1	journal	January 2006
Ensemble forecasting Leutbecher, M.; Palmer, T. N. Journal of Computational Physics, Vol. 227, Issue 7 https://doi.org/10.1016/j.jcp.2007.02.014	journal	March 2008
Roots of Ensemble Forecasting Lewis, John M. Monthly Weather Review, Vol. 133, Issue 7 https://doi.org/10.1175/MWR2949.1	journal	July 2005

Cited By (5)

Robust high‐order semi‐implicit backward difference formulas for Navier‐Stokes equations Loy, Kak Choon; Bourgault, Yves International Journal for Numerical Methods in Fluids, Vol. 90, Issue 4 https://doi.org/10.1002/fld.4721	journal	March 2019
An artificial compressibility ensemble algorithm for a stochastic Stokes‐Darcy model with random hydraulic conductivity and interface conditions He, Xiaoming; Jiang, Nan; Qiu, Changxin International Journal for Numerical Methods in Engineering, Vol. 121, Issue 4 https://doi.org/10.1002/nme.6241	journal	November 2019
An efficient, partitioned ensemble algorithm for simulating ensembles of evolutionary MHD flows at low magnetic Reynolds number Jiang, Nan; Schneier, Michael Numerical Methods for Partial Differential Equations, Vol. 34, Issue 6 https://doi.org/10.1002/num.22281	journal	May 2018
A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems Gunzburger, Max; Jiang, Nan; Wang, Zhu Computational Methods in Applied Mathematics, Vol. 19, Issue 3 https://doi.org/10.1515/cmam-2017-0051	journal	July 2019
A second-order time-stepping scheme for simulating ensembles of parameterized flow problems Gunzburger, Max; Jiang, Nan; Wang, Zhu arXiv https://doi.org/10.48550/arxiv.1706.04060	preprint	January 2017

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97 MATHEMATICS AND COMPUTING
mathematics
blended backward differentiation formula
ensemble calculations
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Title: A second-order ensemble method based on a blended backward differentiation formula timestepping scheme for time-dependent Navier-Stokes equations

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