Local preconditioners for steady and unsteady flow applications

Eli Turkel; Veer N. Vatsa

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 39, Issue: 3, page 515-535
  • ISSN: 0764-583X

Abstract

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Preconditioners for hyperbolic systems are numerical artifacts to accelerate the convergence to a steady state. In addition, the preconditioner should also be included in the artificial viscosity or upwinding terms to improve the accuracy of the steady state solution. For time dependent problems we use a dual time stepping approach. The preconditioner affects the convergence rate and the accuracy of the subiterations within each physical time step. We consider two types of local preconditioners: Jacobi and low speed preconditioning. We can express the algorithm in several sets of variables while using only the conservation variables for the flux terms. We compare the effect of these various variable sets on the efficiency and accuracy of the scheme.

How to cite

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Turkel, Eli, and Vatsa, Veer N.. "Local preconditioners for steady and unsteady flow applications." ESAIM: Mathematical Modelling and Numerical Analysis 39.3 (2010): 515-535. <http://eudml.org/doc/194273>.

@article{Turkel2010,
abstract = { Preconditioners for hyperbolic systems are numerical artifacts to accelerate the convergence to a steady state. In addition, the preconditioner should also be included in the artificial viscosity or upwinding terms to improve the accuracy of the steady state solution. For time dependent problems we use a dual time stepping approach. The preconditioner affects the convergence rate and the accuracy of the subiterations within each physical time step. We consider two types of local preconditioners: Jacobi and low speed preconditioning. We can express the algorithm in several sets of variables while using only the conservation variables for the flux terms. We compare the effect of these various variable sets on the efficiency and accuracy of the scheme. },
author = {Turkel, Eli, Vatsa, Veer N.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Low Mach; preconditioning; Jacobi; Dual time Step; compressible Navier Stokes.},
language = {eng},
month = {3},
number = {3},
pages = {515-535},
publisher = {EDP Sciences},
title = {Local preconditioners for steady and unsteady flow applications},
url = {http://eudml.org/doc/194273},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Turkel, Eli
AU - Vatsa, Veer N.
TI - Local preconditioners for steady and unsteady flow applications
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 3
SP - 515
EP - 535
AB - Preconditioners for hyperbolic systems are numerical artifacts to accelerate the convergence to a steady state. In addition, the preconditioner should also be included in the artificial viscosity or upwinding terms to improve the accuracy of the steady state solution. For time dependent problems we use a dual time stepping approach. The preconditioner affects the convergence rate and the accuracy of the subiterations within each physical time step. We consider two types of local preconditioners: Jacobi and low speed preconditioning. We can express the algorithm in several sets of variables while using only the conservation variables for the flux terms. We compare the effect of these various variable sets on the efficiency and accuracy of the scheme.
LA - eng
KW - Low Mach; preconditioning; Jacobi; Dual time Step; compressible Navier Stokes.
UR - http://eudml.org/doc/194273
ER -

References

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  1. S. Abarbanel and D. Gottlieb, Time splitting for two and three-dimensional Navier-Stokes equations with mixed derivatives. J. Comput. Phys.41 (1981) 1–33.  
  2. S. Allmaras, Analysis of a Local Matrix Preconditioner for the 2-D Navier-Stokes Equations. AIAA Paper 1993-3330 (1993).  
  3. A. Brandt, Multi-level adaptive solutions to boundary value problems. Math. Comp.31 (1977) 333–390.  
  4. D.A. Caughey and A. Jameson, Fast Preconditioned Multigrid Solution of the Euler and Navier-Stokes Equations for Steady Compressible Flows. AIAA Paper 2002-0963 (2002).  
  5. K. Hosseini and J.J. Alonso, Practical Implementation and Improvement of Preconditioning Methods for Explicit Multistage Flow Solvers. AIAA Paper 2004-0763 (2004).  
  6. A. Jameson, The Evolution of Computational Methods in Aerodynamics. ASME J. Appl. Mech.50 (1983) 1052–1070.  
  7. A. Jameson, Time Dependent Calculations Using Multigrid, with Applications to Unsteady Flows past Airfoils and Wings. AIAA Paper 1991-1596 (1991).  
  8. A. Jameson and D.A. Caughey, How Many Steps are Required to Solve the Euler equations of Steady, Compressible Flow: In Search of a Fast Solution Algorithm. AIAA Paper 2001-2673 (2001).  
  9. A. Jameson, W. Schmidt and E. Turkel, Numerical Solutions of the Euler Equations by a Finite Volume Method using Runge-Kutta Time-Stepping Schemes. AIAA Paper 1981-1259 (1981).  
  10. L. Martinelli and A. Jameson, Validation of a Multigrid Method for the Reynolds Averaged Equations. AIAA Paper 1988-0414 (1988).  
  11. N.D. Melson and M.D. Sanetrik, Multigrid Acceleration of Time-Accurate Navier-Stokes Calculations, in 7th Copper Mountain Conference on Multigrid Methods (1995).  
  12. S.A. Pandya, S. Venkateswaran and T.H. Pulliam, Implementation of Preconditioned Dual-Time Procedures in OVERFLOW. AIAA paper 2003-0072 (2003).  
  13. N.A. Pierce and M.B. Giles, Preconditioned multigrid methods for compressible flow codes on stretched meshes. J. Comput. Phys.136 (1997) 425–445.  
  14. J.S. Shuen, K.H. Chen and Y.H. Choi, A Time-Accurate Algorithm for Chemical Non-Equilibrium Viscous Flows at All Speeds. AIAA Paper 1992-3639 (1992).  
  15. R.C. Swanson and E. Turkel, On central difference and upwind schemes. J. Comput. Phys.101 (1992) 292–306.  
  16. E. Turkel, Preconditioned methods for solving the incompressible and low speed compressible equations. J. Comput. Phys.72 (1987) 277–298.  
  17. E. Turkel, A review of preconditioning methods for fluid dynamics. Appl. Numer. Math.12 (1993) 257–284.  
  18. E. Turkel, Preconditioning-Squared Methods for Multidimensional Aerodynamics. AIAA Paper 1997-2025 (1997).  
  19. E. Turkel, Preconditioning Techniques in Computational Fluid Dynamics. An. Rev. Fluid Mech.31 (1999) 385–416.  
  20. E. Turkel, Robust Preconditioning for Steady and Unsteady Viscous Flows. AIAA Paper 2002-0962 (2002).  
  21. E. Turkel and V.N. Vatsa, Effect of artificial viscosity on three-dimensional flow solutions. AIAA Journal32 (1993) 39–45.  
  22. E. Turkel and V.N. Vatsa, Choice of Variables and Preconditioning for Time Dependent Problems. AIAA Paper 2003-3692 (2003).  
  23. E. Turkel, A. Fiterman and B. van Leer, Preconditioning and the Limit to the Incompressible Flow Equations, in Computing the Future: Frontiers of Computational Fluid Dynamics 1994, D.A. Caughey and M.M. Hafez Eds., Wiley Publishing (1994) 215–234.  
  24. E. Turkel, V.N. Vatsa and R. Radespiel, Preconditioning Methods for Low Speed Flow. AIAA Paper 1996-2460 (1996).  
  25. E. Turkel, V.N. Vatsa and V. Venkatakrishnan, Uni-directional Implicit Acceleration Techniques. 14th AIAA Computational Fluid Dynamics Conference. AIAA paper 1999-3265 (1999).  
  26. B. van Leer, W.T. Lee and P.L. Roe, Characteristic Time-Stepping or Local Preconditioning of the Euler Equations. AIAA Paper 1991-1552 (1991).  
  27. V.N. Vatsa and B.W. Wedan, Development of a Multigrid Code for 3-d Navier-Stokes Equations and its Application to a Grid-refinement Study. Comput. Fluids18 (1990) 391–403.  
  28. V.N. Vatsa, M.D. Sanetrik and E.B. Parlette, A Multigrid Based Multiblock Flow Solver for Practical Aerodynamic Configurations, in Computing the Future: Frontiers of Computational Fluid Dynamics 1994, D.A. Caughey and M.M. Hafez Eds., Wiley Publishing (1994) 414–447.  
  29. S. Venkateswaran and L. Merkle, Dual Time Stepping and Preconditioning for Unsteady Computations. AIAA Paper 1995-0078 (1995).  
  30. S. Venkateswaran, D. Li and L. Merkle, Influence of Stagnation Regions on Preconditioned Solutions at Low Speeds. AIAA Paper 2003-0435 (2003).  
  31. L.B. Wigton and R.C. Swanson, Variable Coefficient Implicit Residual Smoothing, 12th International Conference on Numerical Methods in Fluid Dynamics (1990).  
  32. J.P. Withington, J.S. Shuen and V. Yang, A Time Accurate, Implicit Method for Chemically Reacting Flows at All Mach Numbers. AIAA Paper 1991-0581 (1991).  

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