Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Minion, M. L.; Saye, R. I.

doi:10.1016/j.jcp.2018.08.054

Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Journal Article · Wed Sep 05 00:00:00 EDT 2018 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2018.08.054· OSTI ID:1477423

Minion, M. L. ^[1]; Saye, R. I. ^[1]

Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Here, this paper compares and contrasts higher-order, semi-implicit temporal integration strategies for the incompressible Navier–Stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations. Particular focus is placed on the imposition of boundary conditions in the semi-implicit formulation, the accurate treatment of the pressure term, and the smoothness of the numerical solution for different formulations. The main result presented here is the formulation and numerical validation of a single-step, semi-implicit method called spectral deferred pressure corrections, which can in theory obtain arbitrary formal order of accuracy in time. Lastly, numerical results demonstrate up to eighth-order accuracy, scenarios where optimal temporal accuracy is attained, and scenarios where order reduction is observed due to time-dependent boundary conditions.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: AC02-05CH11231

OSTI ID:: 1477423

Alternate ID(s):: OSTI ID: 1635883
OSTI ID: 23078170

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 375; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (58)

Gauge finite element method for incompressible flows E., Weinan; Liu, Jian-Guo International Journal for Numerical Methods in Fluids, Vol. 34, Issue 8 https://doi.org/10.1002/1097-0363(20001230)34:8<701::AID-FLD76>3.0.CO;2-B	journal	January 2000
On pressure boundary conditions for the incompressible Navier-Stokes equations Gresho, Philip M.; Sani, Robert L. International Journal for Numerical Methods in Fluids, Vol. 7, Issue 10 https://doi.org/10.1002/fld.1650071008	journal	October 1987
Finite Difference Schemes for Incompressible Flows in the Velocity–Impulse Density Formulation E., Weinan; Liu, Jian-Guo Journal of Computational Physics, Vol. 130, Issue 1 https://doi.org/10.1006/jcph.1996.5537	journal	January 1997
A Projection Method for Low Speed Flows Colella, Phillip; Pao, Karen Journal of Computational Physics, Vol. 149, Issue 2 https://doi.org/10.1006/jcph.1998.6152	journal	March 1999
Accurate Projection Methods for the Incompressible Navier–Stokes Equations Brown, David L.; Cortez, Ricardo; Minion, Michael L. Journal of Computational Physics, Vol. 168, Issue 2 https://doi.org/10.1006/jcph.2001.6715	journal	April 2001
Iterated deferred corrections for nonlinear operator equations Pereyra, Victor Numerische Mathematik, Vol. 10, Issue 4 https://doi.org/10.1007/BF02162030	journal	November 1967
On improving an approximate solution of a functional equation by deferred corrections Pereyra, Victor Numerische Mathematik, Vol. 8, Issue 4 https://doi.org/10.1007/BF02162981	journal	June 1966
The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods Hairer, Ernst; Roche, Michel; Lubich, Christian Lecture Notes in Mathematics https://doi.org/10.1007/BFb0093947	book	January 1989
Stability of a Cartesian grid projection method for zero Froude number shallow water flows Vater, Stefan; Klein, Rupert Numerische Mathematik, Vol. 113, Issue 1 https://doi.org/10.1007/s00211-009-0224-8	journal	April 2009
Runge-Kutta methods without order reduction for linear initial boundary value problems Alonso-Mallo, Isaías Numerische Mathematik, Vol. 91, Issue 4 https://doi.org/10.1007/s002110100332	journal	June 2002
A class of fully third-order accurate projection methods for solving the incompressible Navier-Stokes equations Ren, Yuxin; Jiang, Yuxi; Liu, Miao'er Acta Mechanica Sinica, Vol. 21, Issue 6 https://doi.org/10.1007/s10409-005-0074-2	journal	December 2005
Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations Layton, Anita T.; Minion, Michael L. BIT Numerical Mathematics, Vol. 45, Issue 2 https://doi.org/10.1007/s10543-005-0016-1	journal	June 2005
Faster SDC convergence on non-equidistant grids by DIRK sweeps Weiser, Martin BIT Numerical Mathematics, Vol. 55, Issue 4 https://doi.org/10.1007/s10543-014-0540-y	journal	December 2014
Partitioned and Implicit–Explicit General Linear Methods for Ordinary Differential Equations Zhang, Hong; Sandu, Adrian; Blaise, Sebastien Journal of Scientific Computing, Vol. 61, Issue 1 https://doi.org/10.1007/s10915-014-9819-z	journal	January 2014
A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows Goda, Katuhiko Journal of Computational Physics, Vol. 30, Issue 1 https://doi.org/10.1016/0021-9991(79)90088-3	journal	January 1979
Application of a fractional-step method to incompressible Navier-Stokes equations Kim, J.; Moin, P. Journal of Computational Physics, Vol. 59, Issue 2 https://doi.org/10.1016/0021-9991(85)90148-2	journal	June 1985
A second-order projection method for the incompressible navier-stokes equations Bell, John B.; Colella, Phillip; Glaz, Harland M. Journal of Computational Physics, Vol. 85, Issue 2 https://doi.org/10.1016/0021-9991(89)90151-4	journal	December 1989
A second-order projection method for variable-density flows Bell, John B.; Marcus, Daniel L. Journal of Computational Physics, Vol. 101, Issue 2 https://doi.org/10.1016/0021-9991(92)90011-M	journal	August 1992
Optimal orders of convergence for Runge–Kutta methods and linear, initial boundary value problems Alonso-Mallo, I.; Palencia, C. Applied Numerical Mathematics, Vol. 44, Issue 1-2 https://doi.org/10.1016/S0168-9274(02)00110-1	journal	January 2003
Additive Runge–Kutta schemes for convection–diffusion–reaction equations Kennedy, Christopher A.; Carpenter, Mark H. Applied Numerical Mathematics, Vol. 44, Issue 1-2 https://doi.org/10.1016/S0168-9274(02)00138-1	journal	January 2003
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations Ascher, Uri M.; Ruuth, Steven J.; Spiteri, Raymond J. Applied Numerical Mathematics, Vol. 25, Issue 2-3 https://doi.org/10.1016/S0168-9274(97)00056-1	journal	November 1997
On the stability of implicit-explicit linear multistep methods Frank, J.; Hundsdorfer, W.; Verwer, J. G. Applied Numerical Mathematics, Vol. 25, Issue 2-3 https://doi.org/10.1016/S0168-9274(97)00059-7	journal	November 1997
Semi-implicit projection methods for incompressible flow based on spectral deferred corrections Minion, Michael L. Applied Numerical Mathematics, Vol. 48, Issue 3-4 https://doi.org/10.1016/j.apnum.2003.11.005	journal	March 2004
An analysis of the spectrum of the discontinuous Galerkin method Krivodonova, Lilia; Qin, Ruibin Applied Numerical Mathematics, Vol. 64 https://doi.org/10.1016/j.apnum.2012.07.008	journal	February 2013
An overview of projection methods for incompressible flows Guermond, J. L.; Minev, P.; Shen, Jie Computer Methods in Applied Mechanics and Engineering, Vol. 195, Issue 44-47 https://doi.org/10.1016/j.cma.2005.10.010	journal	September 2006
Accelerating the convergence of spectral deferred correction methods Huang, Jingfang; Jia, Jun; Minion, Michael Journal of Computational Physics, Vol. 214, Issue 2 https://doi.org/10.1016/j.jcp.2005.10.004	journal	May 2006
A fourth-order auxiliary variable projection method for zero-Mach number gas dynamics Kadioglu, Samet Y.; Klein, Rupert; Minion, Michael L. Journal of Computational Physics, Vol. 227, Issue 3 https://doi.org/10.1016/j.jcp.2007.10.008	journal	January 2008
An accurate and efficient method for the incompressible Navier–Stokes equations using the projection method as a preconditioner Griffith, Boyce E. Journal of Computational Physics, Vol. 228, Issue 20 https://doi.org/10.1016/j.jcp.2009.07.001	journal	November 2009
Second kind integral equation formulation for the modified biharmonic equation and its applications Jiang, Shidong; Kropinski, Mary Catherine A.; Quaife, Bryan D. Journal of Computational Physics, Vol. 249 https://doi.org/10.1016/j.jcp.2013.04.034	journal	September 2013
Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part I Saye, Robert Journal of Computational Physics, Vol. 344 https://doi.org/10.1016/j.jcp.2017.04.076	journal	September 2017
Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part II Saye, Robert Journal of Computational Physics, Vol. 344 https://doi.org/10.1016/j.jcp.2017.05.003	journal	September 2017
Impulse formulation of the Euler equations: general properties and numerical methods Russo, Giovanni; Smereka, Peter Journal of Fluid Mechanics, Vol. 391 https://doi.org/10.1017/S0022112099005170	journal	July 1999
Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems Cockburn, Bernardo; Shu, Chi-Wang Journal of Scientific Computing, Vol. 16, Issue 3, p. 173-261 https://doi.org/10.1023/A:1012873910884	journal	September 2001
Spectral Deferred Correction Methods for Ordinary Differential Equations Dutt, Alok; Greengard, Leslie; Rokhlin, Vladimir Bit Numerical Mathematics, Vol. 40, Issue 2, p. 241-266 https://doi.org/10.1023/A:1022338906936	journal	June 2000
Motion of Halley's comet during the return of 1910 Zadunaisky, Pedro E. The Astronomical Journal, Vol. 71 https://doi.org/10.1086/109848	journal	February 1966
Maestro: an Adaptive low mach Number Hydrodynamics Algorithm for Stellar Flows Nonaka, A.; Almgren, A. S.; Bell, J. B. The Astrophysical Journal Supplement Series, Vol. 188, Issue 2 https://doi.org/10.1088/0067-0049/188/2/358	journal	May 2010
Vortex Methods: Theory and Practice Koumoutsakos, G-H Cottet and P. D. Measurement Science and Technology, Vol. 12, Issue 3 https://doi.org/10.1088/0957-0233/12/3/704	journal	February 2001
Numerical simulation of laminar reacting flows with complex chemistry Day, M. S.; Bell, J. B. Combustion Theory and Modelling, Vol. 4, Issue 4 https://doi.org/10.1088/1364-7830/4/4/309	journal	December 2000
Projection method III: Spatial discretization on the staggered grid E., Weinan; Liu, Jian-Guo Mathematics of Computation, Vol. 71, Issue 237 https://doi.org/10.1090/S0025-5718-01-01313-8	journal	May 2001
Avoiding the order reduction of Runge-Kutta methods for linear initial boundary value problems Calvo, M. P.; Palencia, C. Mathematics of Computation, Vol. 71, Issue 240 https://doi.org/10.1090/S0025-5718-01-01362-X	journal	November 2001
Numerical solution of the Navier-Stokes equations Chorin, Alexandre Joel Mathematics of Computation, Vol. 22, Issue 104 https://doi.org/10.1090/S0025-5718-1968-0242392-2	journal	January 1968
On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations Prothero, A.; Robinson, A. Mathematics of Computation, Vol. 28, Issue 125 https://doi.org/10.1090/S0025-5718-1974-0331793-2	journal	January 1974
Implicit–explicit multistep methods for nonlinear parabolic equations Akrivis, Georgios Mathematics of Computation, Vol. 82, Issue 281 https://doi.org/10.1090/S0025-5718-2012-02628-7	journal	July 2012
Interfacial gauge methods for incompressible fluid dynamics Saye, Robert Science Advances, Vol. 2, Issue 6 https://doi.org/10.1126/sciadv.1501869	journal	June 2016
Error Analysis of IMEX Runge–Kutta Methods Derived from Differential-Algebraic Systems Boscarino, Sebastiano SIAM Journal on Numerical Analysis, Vol. 45, Issue 4 https://doi.org/10.1137/060656929	journal	January 2007
Order Results for Implicit Runge–Kutta Methods Applied to Stiff Systems Frank, Reinhard; Schneid, Josef; Ueberhuber, Christoph W. SIAM Journal on Numerical Analysis, Vol. 22, Issue 3 https://doi.org/10.1137/0722031	journal	June 1985
Implicit-Explicit Methods for Time-Dependent Partial Differential Equations Ascher, Uri M.; Ruuth, Steven J.; Wetton, Brian T. R. SIAM Journal on Numerical Analysis, Vol. 32, Issue 3 https://doi.org/10.1137/0732037	journal	June 1995
On a Class of Uniformly Accurate IMEX Runge–Kutta Schemes and Applications to Hyperbolic Systems with Relaxation Boscarino, Sebastiano; Russo, Giovanni SIAM Journal on Scientific Computing, Vol. 31, Issue 3 https://doi.org/10.1137/080713562	journal	January 2009
Extrapolated Implicit-Explicit Time Stepping Constantinescu, Emil M.; Sandu, Adrian SIAM Journal on Scientific Computing, Vol. 31, Issue 6 https://doi.org/10.1137/080732833	journal	January 2010
The Theoretical Accuracy of Runge–Kutta Time Discretizations for the Initial Boundary Value Problem: A Study of the Boundary Error Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul SIAM Journal on Scientific Computing, Vol. 16, Issue 6 https://doi.org/10.1137/0916072	journal	November 1995
On the Use of Higher-Order Projection Methods for Incompressible Turbulent Flow Almgren, A. S.; Aspden, A. J.; Bell, J. B. SIAM Journal on Scientific Computing, Vol. 35, Issue 1 https://doi.org/10.1137/110829386	journal	January 2013
A Helmholtz--Hodge Projection Method Using an Iterative Gauge Computation to Solve the 3D Generalized Stokes Problem Garba, Abdou; Haldenwang, Pierre SIAM Journal on Scientific Computing, Vol. 35, Issue 3 https://doi.org/10.1137/110860902	journal	January 2013
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems Arnold, Douglas N.; Brezzi, Franco; Cockburn, Bernardo SIAM Journal on Numerical Analysis, Vol. 39, Issue 5 https://doi.org/10.1137/S0036142901384162	journal	January 2002
The Correct Formulation of Intermediate Boundary Conditions for Runge--Kutta Time Integration of Initial Boundary Value Problems Pathria, D. SIAM Journal on Scientific Computing, Vol. 18, Issue 5 https://doi.org/10.1137/S1064827594273948	journal	September 1997
Efficient Variable-Coefficient Finite-Volume Stokes Solvers Cai, Mingchao; Nonaka, Andy; Bell, John B. Communications in Computational Physics, Vol. 16, Issue 5 https://doi.org/10.4208/cicp.070114.170614a	journal	November 2014
On the Gauss Runge-Kutta and Method of Lines Transpose for Initial-Boundary Value Parabolic PDEs Zhang, Bo; Chen, Dangxing; Huang, Jingfang Communications in Computational Physics, Vol. 24, Issue 4 https://doi.org/10.4208/cicp.2018.hh80.09	journal	January 2018
Gauge Method for Viscous Incompressible Flows E., Weinan; Liu, Jian-Guo Communications in Mathematical Sciences, Vol. 1, Issue 2 https://doi.org/10.4310/CMS.2003.v1.n2.a6	journal	January 2003
Semi-implicit spectral deferred correction methods for ordinary differential equations Minion, Michael L. Communications in Mathematical Sciences, Vol. 1, Issue 3 https://doi.org/10.4310/CMS.2003.v1.n3.a6	journal	January 2003

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