Jemcov, Aleksandar; Maruszewski, Joseph P Algorithm stabilization and acceleration in computational fluid dynamics: exploiting recursive properties of fixed point algorithms Journal Article Computational Fluid Dynamics and Heat Transfer: Emerging Topics, 23 , pp. 459, 2011. Abstract | Links | BibTeX | Tags: Acceleration, Linear Solver @article{jemcov201112,
title = {Algorithm stabilization and acceleration in computational fluid dynamics: exploiting recursive properties of fixed point algorithms},
author = { Aleksandar Jemcov and Joseph P Maruszewski},
doi = {10.2495/978-1-84564-144-3/12 },
year = {2011},
date = {2011-01-01},
journal = {Computational Fluid Dynamics and Heat Transfer: Emerging Topics},
volume = {23},
pages = {459},
publisher = {WIT Press},
abstract = {Convergence acceleration of nonlinear flow solvers through use of techniques that exploit recursive properties fixed-point methods of computational fluid dynamics (CFD) algorithms is presented.
It is shown that nonlinear iterative algorithms create Krylov subspace that can be considered linear within the second order of accuracy in the vicinity of the local solutions.
Moreover, repeated application of the fixed-point iterations give rise to various vector extrapolation methods that are used to accelerate underlying iterative method.
In particular, suitability of the reduced rank extrapolation (RRE) algorithm for the use of convergence acceleration of nonlinear flow solvers is examined.
In the RRE algorithm, the solution is obtained through a linear combination of Krylov vectors with weighting coefficients obtained by minimizing L2 norm of error in this space with properly chosen constraint conditions.
This process effectively defines vector sequence extrapolation process in Krylov subspace that corresponds to the GMRES method applied to nonlinear problems.
Moreover, when the RRE algorithm is used to solve nonlinear problems, the flow solver plays the role of the preconditioner for the nonlinear GMRES method.
Benefits of the application of the RRE algorithm include better convergence rates, removal of residual stalling, and improved coupling between equations in numerical models.
Proposed algorithm is independent of the type of flow solver and it is equally applicable to explicit-, implicit-, pressure-, and density-, based algorithms.
RRE algorithm can also be considered as an generalization of matrix-free algorithms that are used extensively in CFD.},
keywords = {Acceleration, Linear Solver},
pubstate = {published},
tppubtype = {article}
}
Convergence acceleration of nonlinear flow solvers through use of techniques that exploit recursive properties fixed-point methods of computational fluid dynamics (CFD) algorithms is presented.
It is shown that nonlinear iterative algorithms create Krylov subspace that can be considered linear within the second order of accuracy in the vicinity of the local solutions.
Moreover, repeated application of the fixed-point iterations give rise to various vector extrapolation methods that are used to accelerate underlying iterative method.
In particular, suitability of the reduced rank extrapolation (RRE) algorithm for the use of convergence acceleration of nonlinear flow solvers is examined.
In the RRE algorithm, the solution is obtained through a linear combination of Krylov vectors with weighting coefficients obtained by minimizing L2 norm of error in this space with properly chosen constraint conditions.
This process effectively defines vector sequence extrapolation process in Krylov subspace that corresponds to the GMRES method applied to nonlinear problems.
Moreover, when the RRE algorithm is used to solve nonlinear problems, the flow solver plays the role of the preconditioner for the nonlinear GMRES method.
Benefits of the application of the RRE algorithm include better convergence rates, removal of residual stalling, and improved coupling between equations in numerical models.
Proposed algorithm is independent of the type of flow solver and it is equally applicable to explicit-, implicit-, pressure-, and density-, based algorithms.
RRE algorithm can also be considered as an generalization of matrix-free algorithms that are used extensively in CFD. |

Jemcov, Aleksandar; Maruszewski, Joseph P Nonlinear Flow Solver Acceleration by Reduced Rank Extrapolation Incollection 46th AIAA Aerospace Sciences Meeting and Exhibit, pp. 609, 2008. Abstract | Links | BibTeX | Tags: Acceleration, Algorithm, Extrapolation, GMRES, Nonlinear, RRE, SIMPLE, Solver @incollection{jemcov2008nonlinear,
title = {Nonlinear Flow Solver Acceleration by Reduced Rank Extrapolation},
author = { Aleksandar Jemcov and Joseph P Maruszewski},
url = {https://www.researchgate.net/profile/Aleksandar_Jemcov/publication/275275199_Nonlinear_Flow_Solver_Acceleration_by_Reduced_Rank_Extrapolation/links/5536ca190cf2058efdea9401.pdf?origin=publication_detail_rebranded&ev=pub_int_prw_xdl&msrp=mHcz4Kt3tcM0fKKvzoZAfJu5CZ7bJAZTebm7svkdI8HyTV2DqiloQBxWK3ZC9HHtc1VaKywIo53Gq4ACRkkbmQ%3D%3D_mZOVHfxww2SuKoJv0YEv1yJ19XXhbwqiKJ%2FuZ%2FsAeTe%2BZ2RdfWruA0KDkoFLyPO0ebx%2BKC9jrGwRj%2B%2F%2BKjsf7A%3D%3D},
year = {2008},
date = {2008-01-01},
booktitle = {46th AIAA Aerospace Sciences Meeting and Exhibit},
pages = {609},
abstract = {Convergence acceleration of nonlinear flow solvers through use of vector sequence ex-trapolation techniques is presented. In particular, suitability of the Reduced Rank Extrap-olation (RRE) algorithm for the use of convergence acceleration of nonlinear flow solvers is examined. In the RRE algorithm, the solution is obtained through a linear combination of Krylov vectors with weighting coefficients obtained by minimizing L2 norm of error in this space with properly chosen constraint conditions. This process effectively defines vector sequence extrapolation process in Krylov subspace that corresponds to the GMRES method applied to nonlinear problems. Moreover, when the RRE algorithm is used to solve nonlinear problems, the flow solver plays the role of the preconditioner for the non-linear GMRES method. Benefits of the application of the RRE algorithm include better convergence rates, removal of residual stalling and improved coupling between equations in numerical models. Proposed algorithm is independent of the type of flow solver and it is equally applicable to explicit, implicit, pressure and density based algorithms. Nomenclature Q Vector of conserved variables R Residual vector F c Vector of convective fluxes F v Vector of viscous fluxes S Source vector H Total enthalpy, J/m 3 p pressure, Pa τ ij viscous tensor ρ Density, Kg/m 3 u Velocity vector, m/s u X-component of velocity vector, m/s v Y-component of velocity vector, m/s w Z-component of velocity vector, m/s V Contravariant velocity, m/s f e,i Vector of external forces Φ Generic transport variable M Nonlinear preconditioning operator F Fixed-point function ∂ Q (·) Jacobian with respect to Q α ν Extrapolation coefficients c ν o t Time, s ∂Ω Boundary of computational domain Ω Computational domain},
keywords = {Acceleration, Algorithm, Extrapolation, GMRES, Nonlinear, RRE, SIMPLE, Solver},
pubstate = {published},
tppubtype = {incollection}
}
Convergence acceleration of nonlinear flow solvers through use of vector sequence ex-trapolation techniques is presented. In particular, suitability of the Reduced Rank Extrap-olation (RRE) algorithm for the use of convergence acceleration of nonlinear flow solvers is examined. In the RRE algorithm, the solution is obtained through a linear combination of Krylov vectors with weighting coefficients obtained by minimizing L2 norm of error in this space with properly chosen constraint conditions. This process effectively defines vector sequence extrapolation process in Krylov subspace that corresponds to the GMRES method applied to nonlinear problems. Moreover, when the RRE algorithm is used to solve nonlinear problems, the flow solver plays the role of the preconditioner for the non-linear GMRES method. Benefits of the application of the RRE algorithm include better convergence rates, removal of residual stalling and improved coupling between equations in numerical models. Proposed algorithm is independent of the type of flow solver and it is equally applicable to explicit, implicit, pressure and density based algorithms. Nomenclature Q Vector of conserved variables R Residual vector F c Vector of convective fluxes F v Vector of viscous fluxes S Source vector H Total enthalpy, J/m 3 p pressure, Pa τ ij viscous tensor ρ Density, Kg/m 3 u Velocity vector, m/s u X-component of velocity vector, m/s v Y-component of velocity vector, m/s w Z-component of velocity vector, m/s V Contravariant velocity, m/s f e,i Vector of external forces Φ Generic transport variable M Nonlinear preconditioning operator F Fixed-point function ∂ Q (·) Jacobian with respect to Q α ν Extrapolation coefficients c ν o t Time, s ∂Ω Boundary of computational domain Ω Computational domain |