DOI QR코드

DOI QR Code

A PARALLEL IMPLEMENTATION OF A RELAXED HSS PRECONDITIONER FOR SADDLE POINT PROBLEMS FROM THE NAVIER-STOKES EQUATIONS

  • JANG, HO-JONG (DEPARTMENT OF MATHEMATICS, RESEARCH INSTITUTE FOR NATURAL SCIENCES, HANYANG UNIVERSITY) ;
  • YOUN, KIHANG (DEPARTMENT OF MATHEMATICS, RESEARCH INSTITUTE FOR NATURAL SCIENCES, HANYANG UNIVERSITY)
  • Received : 2018.05.29
  • Accepted : 2018.09.10
  • Published : 2018.09.25

Abstract

We describe a parallel implementation of a relaxed Hermitian and skew-Hermitian splitting preconditioner for the numerical solution of saddle point problems arising from the steady incompressible Navier-Stokes equations. The equations are linearized by the Picard iteration and discretized with the finite element and finite difference schemes on two-dimensional and three-dimensional domains. We report strong scalability results for up to 32 cores.

References

  1. Z. -Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), 603-626. https://doi.org/10.1137/S0895479801395458
  2. S. Balay, S. Abhyankar, F. Adams, J. Brown and P. Brune et al., PETSc Users Manual, Argonne National Laboratory, ANL-95/11 - Revision 3.9 (2018).
  3. M. Benzi and G. H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), 20-41. https://doi.org/10.1137/S0895479802417106
  4. M. Benzi, G. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numer, 14 (2005), 1-137. https://doi.org/10.1017/S0962492904000212
  5. M. Benzi and J. Liu, An efficient solver for the incompressible Navier-Stokes equations in rotation form, SIAM J. Sci. Comput., 29 (2007), 1959-1981. https://doi.org/10.1137/060658825
  6. M. Benzi and Z. Wang, A parallel implementation of the modified augmented Lagrangian preconditioner for the incompressible Navier-Stokes equations, Numer Algor, 64 (2013), 73-84. https://doi.org/10.1007/s11075-012-9655-x
  7. Y. Cao, L. Q. Yao, M. Q. Jiang and Q. Niu, A relaxed HSS preconditioner for saddle point problems from meshfree discretization, J. Comput. Math, 31 (2013), 398-421. https://doi.org/10.4208/jcm.1304-m4209
  8. H. C. Elman, A. Ramage and D. J. Silvester, IFISS: a Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Software, 33 (2007), Article 14.
  9. H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Oxford University Press, Oxford, UK, 2005.
  10. F. H. Harlow, J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8(1965), 2182-2189. https://doi.org/10.1063/1.1761178
  11. M. H. Salkuyeh and M. Masoudi, A new relaxed HSS preconditioner for saddle point problems, Numer. Algor., 74 (2017), 781-795. https://doi.org/10.1007/s11075-016-0171-2