DOI: https://doi.org/10.26089/NumMet.v18r107

On combining the techniques for convergence acceleration of iteration processes during the numerical solution of Navier-Stokes equations

Authors

  • E.V. Vorozhtsov
  • V.P. Shapeev

Keywords:

preconditioning
Krylov subspaces
multigrid algorithms
Gauss–Seidel iterations
Navier–Stokes equations
the method of collocations and least residuals

Abstract

The problem of accelerating the iteration process of the numerical solution of boundary value problems for partial differential equations by the method of collocations and least residuals (CLR) is considered. In the CLR method it is proposed to simultaneously apply three techniques for accelerating the iteration process: a preconditioner, a multigrid algorithm, and the Krylov method. A two-parameter preconditioner is studied. It is proposed to find the optimal values of its parameters by the numerical solution of a relatively computationally inexpensive problem of minimizing the condition number of the system of linear algebraic equations for the approximate problem. The use of the found preconditioner substantially speeds up the iteration process. The individual effect of each technique as well as the effect of their combined use on accelerating the entire iteration process acceleration are analyzed. The application of the algorithm based on the Krylov subspaces gives the most significant contribution. A simultaneous combined use of all the three techniques for accelerating the iteration process of solving the boundary value problems for the two-dimensional Navier-Stokes equations reduces the CPU time of their solution by a factor of up to 160 compared to the case when no such technique is applied. The proposed combination of the above techniques for accelerating the iteration processes may also be implemented in the framework of other numerical methods for solving the partial differential equations.

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Published

2017-03-06

Issue

Vol. 18 (2017): Issue 1.

Section

Section 1. Numerical methods and applications

Author Biographies

E.V. Vorozhtsov

S.A. Khristianovich Institute of Theoretical and Applied Mechanics of SB RAS
• Leading Researcher

V.P. Shapeev

S.A. Khristianovich Institute of Theoretical and Applied Mechanics of SB RAS
• Chief Researcher

References

  1. W. S. Edwards, L. S. Tuckerman, R. A. Friesner, and D. C. Sorensen, “Krylov Methods for the Incompressible Navier-Stokes Equations,” J. Comput. Phys. 110 (1), 82-102 (1994).
  2. D. A. Knoll and D. E. Keyes, “Jacobian-Free Newton-Krylov Methods: A Survey of Approaches and Applications,” J. Comput. Phys. 193 (2), 357-397 (2004).
  3. B. E. Griffith, “An Accurate and Efficient Method for the Incompressible Navier-Stokes Equations Using the Projection Method as a Preconditioner,” J. Comput. Phys. 228 (20), 7565-7595 (2009).
  4. Y. Saad, Numerical Methods for Large Eigenvalue Problems (Manchester Univ. Press, Manchester, 1991).
  5. A. N. Krylov, “On the Numerical Solution of the Equation by Which the Frequency of Small Oscillations is Determined in Technical Problems,” Izv. Akad. Nauk SSSR, Ser. Fiz.-Mat., No 4, 491-539 (1931).
  6. A. G. Sleptsov, “On Convergence Acceleration of Linear Iterations, II,” Modelir. Mekhan. 3 (5), 118-125 (1989).
  7. R. P. Fedorenko, “The Speed of Convergence of One Iterative Process,” Zh. Vychisl. Mat. Mat. Fiz. 4 (3), 559-564 (1964) [USSR Comput. Math. Math. Phys. 4 (3), 227-235 (1964)].
  8. J. Piquet and X. Vasseur, “A Nonstandard Multigrid Method with Flexible Multiple Semicoarsening for the Numerical Solution of the Pressure Equation in a Navier-Stokes Solver,” Numer. Algorithms 24 (4), 333-355 (2000).
  9. G. Jothiprasad, D. J. Mavriplis, and D. A. Caughey, “Higher-Order Time Integration Schemes for the Unsteady Navier-Stokes Equations on Unstructured Meshes,” J. Comput. Phys. 191 (2), 542-566 (2003).
  10. L. Ge and F. Sotiropoulos, “A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries,” J. Comput. Phys. 225 (2), 1782-1809 (2007).
  11. P. Lucas, A. H. van Zuijlen, and H. Bijl, “Fast Unsteady Flow Computations with a Jacobian-Free Newton-Krylov Algorithm,” J. Comput. Phys. 229 (24), 9201-9215 (2010).
  12. M. M. Nasr-Azadani and E. Meiburg, “TURBINS: An Immersed Boundary, Navier-Stokes Code for the Simulation of Gravity and Turbidity Currents Interacting with Complex Topographies,” Comput. Fluids 45 (1), 14-28 (2011).
  13. M. Wang and L. Chen, “Multigrid Methods for the Stokes Equations using Distributive Gauss-Seidel Relaxations Based on the Least Squares Commutator,” J. Sci. Comput. 56 (2), 409-431 (2013).
  14. M. Nickaeen, A. Ouazzi, and S. Turek, “Newton Multigrid Least-Squares FEM for the V-V-P Formulation of the Navier-Stokes Equations,” J. Comput. Phys. 256, 416-427 (2014).
  15. F. A. Fairag and A. J. Wathen, “A Block Preconditioning Technique for the Streamfunction-Vorticity Formulation of the Navier-Stokes Equations,” Numer. Methods Partial Differ. Equ. 28 (3), 888-898 (2012).
  16. M. Benzi and Z. Wang, “Analysis of Augmented Lagrangian-Based Preconditioners for the Steady Incompressible Navier-Stokes Equations,” SIAM J. Sci. Comput. 33 (5), 2761-2784 (2011).
  17. B. N. Jiang, T. L. Lin, and L. A. Povinelli, “Large-Scale Computation of Incompressible Viscous Flow by Least-Squares Finite Element Method,” Comput. Meth. Appl. Mech. Eng. 114 (3-4), 213-231 (1994).
  18. M. Ramšak and L. Škerget, “A Subdomain Boundary Element Method for High-Reynolds Laminar Flow using Stream Function-Vorticity Formulation,” Int. J. Numer. Meth. Fluids 46 (8), 815-847 (2004).
  19. A. V. Plyasunova and A. G. Sleptsov, “Collocation Grid Method for Solving Nonlinear Parabolic Equations on Moving Grids,” Model. Mekh. 18 (4), 116-137 (1987).
  20. G. F. Carey and B. N. Jiang, “Least-Squares Finite Element Method and Preconditioned Conjugate Gradient Solution,” Int. J. Numer. Methods Eng. 24 (7), 1283-1296 (1987).
  21. B. N. Jiang, The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics (Springer, Berlin, 1998).
  22. P. B. Bochev and M. D. Gunzburger, “Finite Element Methods of Least-Squares Type,” SIAM Rev. 40 (4), 789-837 (1998).
  23. B. F. Soares, R. V. Garcia, P. C. Pinto, and E. C. Romao, “Interval Study of Convergence in the Solution of 1D Burgers by Least Squares Finite Element Method (LSFEM) + Newton Linearization,” Sci. Res. Essays 10 (16), 522-530 (2015).
  24. L. G. Semin, A. G. Sleptsov, and V. P. Shapeev, “Collocation and Least Squares Method for Stokes Equations,” Vychisl. Tekhnol. 1 (2), 90-98 (1996).
  25. L. Semin and V. Shapeev, “Constructing the Numerical Method for Navier-Stokes Equations Using Computer Algebra System,” in Lecture Notes in Computer Science (Springer, Heidelberg, 2005), Vol. 3718, pp. 367-378.
  26. V. I. Isaev and V. P. Shapeev, “Development of the Collocations and Least Squares Method,” Tr. Inst. Mat. Mekh. UrO RAN 14 (1), 41-60 (2008) [Proc. Steklov Inst. Math. 261 (Suppl. 1), S87-S106 (2008)].
  27. V. I. Isaev, V. P. Shapeev, and A. N. Cherepanov, “Numerical Simulation of Laser Welding of Thin Metallic Plates Taking into Account Convection in the Welding Pool,” Teplofiz. Aeromekh. 17 (3), 451-466 (2010) [Thermophys. Aeromech. 17 (3), 419-434 (2010)].
  28. V. I. Isaev and V. P. Shapeev, “High-Order Accurate Collocations and Least Squares Method for Solving the Navier-Stokes Equations,” Dokl. Akad. Nauk 442 (4), 442-445 (2012) [Dokl. Math. 85 (1), 71-74 (2012)].
  29. A. G. Sleptsov and Yu. I. Shokin, “An Adaptive Grid-Projection Method for Elliptic Problems,” Zh. Vychisl. Mat. Mat. Fiz. 37 (5), 572-586 (1997) [Comput. Math. Math. Phys. 37 (5), 558-571 (1997)].
  30. V. V. Beljaev and V. P. Shapeev, “The Collocation and Least Squares Method on Adaptive Grids in a Domain with a Curvilinear Boundary,” Vychisl. Tekhnol. 5 (4), 13-21 (2000).
  31. V. P. Shapeev, V. I. Isaev, and S. V. Idimeshev, “The Collocations and Least Squares Method: Application to Numerical Solution of the Navier-Stokes Equations,” in Proc. 6th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012), Vienna, Austria, Vienna, September 10-14, 2012 (Vienna Univ. of Tech., Vienna, 2012), p. 359.
  32. V. P. Shapeev and E. V. Vorozhtsov, “Symbolic-Numeric Implementation of the Method of Collocations and Least Squares for 3D Navier-Stokes Equations,” in Lecture Notes in Computer Science (Springer, Heidelberg, 2012), Vol. 7442, pp. 321-333.
  33. V. P. Shapeev, E. V. Vorozhtsov, V. I. Isaev, and S. V. Idimeshev, “The Method of Collocations and Least Residuals for Three-Dimensional Navier-Stokes Equations,” Vychisl. Metody Programm. 14, 306-322 (2013).
  34. V. P. Shapeev and E. V. Vorozhtsov, “CAS Application to the Construction of the Collocations and Least Residuals Method for the Solution of 3D Navier-Stokes Equations,” in Lecture Notes in Computer Science (Springer, Heidelberg, 2013), Vol. 8136, pp. 381-392.
  35. V. Shapeev, “Collocation and Least Residuals Method and Its Applications,” EPJ Web of Conferences 108 (2016).
    doi 10.1051/epjconf/201610801009
  36. V. I. Isaev and V. P. Shapeev, “High-Accuracy Versions of the Collocations and Least Squares Method for the Numerical Solution of the Navier-Stokes Equations,” Zh. Vychisl. Mat. Mat. Fiz. 50 (10), 1758-1770 (2010) [Comput. Math. Math. Phys. 50 (10), 1670-1681 (2010)].
  37. A. V. Shapeev and P. Lin, “An Asymptotic Fitting Finite Element Method with Exponential Mesh Refinement for Accurate Computation of Corner Eddies in Viscous Flows,” SIAM J. Sci. Comput. 31 (3), 1874-1900 (2009).
  38. O. Botella and R. Peyret, “Benchmark Spectral Results on the Lid-Driven Cavity Flow,” Comput. Fluids 27 (4), 421-433 (1998).
  39. S. K. Golushko, S. V. Idimeshev, and V. P. Shapeev, “Application of Collocations and Least Residuals Method to Problems of the Isotropic Plates Theory,” Vychisl. Tekhnol. 18 (6), 31-43 (2013).
  40. S. K. Golushko, S. V. Idimeshev, and V. P. Shapeev, “Development and Application of Collocations and Least Residuals Method to the Solution of Problems in Mechanics of Anisotropic Laminated Plates,” Vychisl. Tekhnol. 19 (5), 24-36 (2014).
  41. V. P. Shapeev and V. A. Belyaev, “Versions of High Order Accuracy Collocation and Least Residuals Method in the Domain with a Curvilinear Boundary,” Vychisl. Tekhnol. 21 (5), 95-110 (2016).
  42. D. Kharenko, C. Padovani, A. Pagni, et al., “Free Longitudinal Vibrations of Bimodular Beams: A Comparative Study,” Int. J. Str. Stab. Dyn. 11 (1), 23-56 (2011).
  43. V. P. Shapeev and E. V. Vorozhtsov, “CAS Application to the Construction of the Collocations and Least Residuals Method for the Solution of the Burgers and Korteweg-de Vries-Burgers Equations,” in Lecture Notes in Computer Science (Springer, Heidelberg, 2014), Vol. 8660, pp. 432-446.
  44. V. I. Isaev, A. N. Cherepanov, and V. P. Shapeev, “Numerical Study of Heat Modes of Laser Welding of Dissimilar Metals with an Intermediate Insert,” Int. J. Heat Mass Transfer 99, 711-720 (2016).
  45. E. V. Vorozhtsov and V. P. Shapeev, “Acceleration of Iterative Processes for Solving Boundary Value Problems by Combining the Krylov and Fedorenko Methods,” Simvol Nauki 10 (2), 24-43 (2015).
  46. V. I. Isaev, V. P. Shapeev, and S. A. Eremin, “An Investigation of the Collocation and the Least Squares Method for Solution of Boundary Value Problems for the Navier-Stokes and Poisson Equations,” Vychisl. Tekhnol. 12 (3), 53-70 (2007).
  47. I. B. Petrov and A. I. Lobanov, Lectures on Computational Mathematics (Binom, Moscow, 2006) [in Russian].
  48. P. H. Chiu, T. W. H. Sheu, and R. K. Lin, “An Effective Explicit Pressure Gradient Scheme Implemented in the Two-Level Non-Staggered Grids for Incompressible Navier-Stokes Equations,” J. Comput. Phys. 227 (8), 4018-4037 (2008).
  49. S. Wolfram, The Mathematica Book (Wolfram Media, Champaign, 2003).
  50. Numerical Analysis Library of Moscow University Research Computing Center, Computing All Eigenvalues of a Symmetric Matrix by the QL Algorithm with Shifts.
    http://num-anal.srcc.msu.ru/lib.na/cat/ae/aeh2r.htm . Cited February 15, 2017.
  51. P. Wesseling, An Introduction to Multigrid Methods (Wiley, Chichester, 1992).
  52. T. W. Tee and I. Sobey, Spectral Method for the Unsteady Incompressible Navier-Stokes Equations in Gauge Formulation , Report NA04/09 (Numerical Analysis Group, Oxford, 2004),
    http://eprints.maths.ox.ac.uk/1181/. Cited February 15, 2017.
  53. S. Albensoeder and H. C. Kuhlmann, “Accurate Three-Dimensional Lid-Driven Cavity Flow,” J. Comput. Phys. 206 (2), 536-558 (2005).
  54. E. Erturk, T. C. Corke, and C. Gökçöl, “Numerical Solutions of 2-D Steady Incompressible Driven Cavity Flow at High Reynolds Numbers,” Int. J. Numer. Methods Fluids 48 (7), 747-774 (2005).
  55. E. Erturk, “Discussions on Driven Cavity Flow,” Int. J. Numer. Methods Fluids 60 (3), 275-294 (2009).
  56. R. Lim and D. Sheen, “Nonconforming Finite Element Method Applied to the Driven Cavity Flow,” arXiv preprint: 1502.04217v3 [math.NA] (Cornell Univ. Library, Ithaca, 2016),
    available at
    https://arxiv.org/abs/1502.04217

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