Higher-order time integration schemes for the unsteady Navier–Stokes equations on unstructured meshes

The efficiency gains obtained using higher-order implicit Runge–Kutta (RK) schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier–Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at eac...

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Published in:	Journal of computational physics Vol. 191; no. 2; pp. 542 - 566
Main Authors:	Jothiprasad, Giridhar, Mavriplis, Dimitri J., Caughey, David A.
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 01.11.2003
Subjects:	Jacobian-free Newton Multigrid Navier–Stokes Runge–Kutta methods Unstructured Jacobian-free Newton Navier–Stokes Unstructured Runge–Kutta methods Multigrid
ISSN:	0021-9991, 1090-2716
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Abstract	The efficiency gains obtained using higher-order implicit Runge–Kutta (RK) schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier–Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each time step are presented. The first algorithm (nonlinear multigrid, NMG) is a pseudo-time-stepping scheme which employs a nonlinear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on inexact Newton’s methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson’s iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the generalized minimal residual method. Results demonstrating the relative superiority of these Newton’s method based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes such as fourth-order Runge–Kutta (RK64) with the more efficient inexact Newton’s method based schemes (LMG).
AbstractList	The efficiency gains obtained using higher-order implicit Runge-Kutta (RK) schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each time step are presented. The first algorithm (nonlinear multigrid, NMG) is a pseudo- time-stepping scheme which employs a nonlinear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the generalized minimal residual method. Results demonstrating the relative superiority of these Newton's method based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes such as fourth- order Runge-Kutta (RK64) with the more efficient inexact Newton's method based schemes (LMG). (Author) The efficiency gains obtained using higher-order implicit Runge–Kutta (RK) schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier–Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each time step are presented. The first algorithm (nonlinear multigrid, NMG) is a pseudo-time-stepping scheme which employs a nonlinear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on inexact Newton’s methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson’s iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the generalized minimal residual method. Results demonstrating the relative superiority of these Newton’s method based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes such as fourth-order Runge–Kutta (RK64) with the more efficient inexact Newton’s method based schemes (LMG).
Author	Mavriplis, Dimitri J. Caughey, David A. Jothiprasad, Giridhar
Author_xml	– sequence: 1 givenname: Giridhar surname: Jothiprasad fullname: Jothiprasad, Giridhar email: gj24@cornell.edu organization: Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA – sequence: 2 givenname: Dimitri J. surname: Mavriplis fullname: Mavriplis, Dimitri J. email: dimitri@icase.edu organization: National Institute of Aerospace, 144 Research Drive, Hampton, VA 23666, USA – sequence: 3 givenname: David A. surname: Caughey fullname: Caughey, David A. email: dac5@cornell.edu organization: Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
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Snippet	The efficiency gains obtained using higher-order implicit Runge–Kutta (RK) schemes as compared with the second-order accurate backward difference schemes for... The efficiency gains obtained using higher-order implicit Runge-Kutta (RK) schemes as compared with the second-order accurate backward difference schemes for...
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Title	Higher-order time integration schemes for the unsteady Navier–Stokes equations on unstructured meshes
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