본문 바로가기
KERD
논문KCI 등재

Analysis of some projection methods for the incompressible fluids with microstructure

2

조회

0

다운로드

0

인용

0

참고문헌

키워드

주제분류수학

초록

In this article, some projection methods (or fractional-step methods) are proposed and analyzed for the micropolar Navier-Stokes equations (MNSE). These methods allow us to decouple the MNSE system into two sub-problems at each timestep, one is the linear and angular velocities system, the other is the pressure system. Both first-order and second-order projection methods are considered. For the classical first-order projection scheme, the stability and error estimates for the linear and angular velocities and the pressure are established rigorously. In addition, a modified first-order projection scheme which leads to some improved error estimates is also proposed and analyzed. We also present the second-order projection method which is unconditionally stable. Ample numerical experiments are performed to confirm the theoretical predictions and demonstrate the efficiency of the methods.

Abstract

In this article, some projection methods (or fractional-step methods) are proposed and analyzed for the micropolar Navier-Stokes equations (MNSE). These methods allow us to decouple the MNSE system into two sub-problems at each timestep, one is the linear and angular velocities system, the other is the pressure system. Both first-order and second-order projection methods are considered. For the classical first-order projection scheme, the stability and error estimates for the linear and angular velocities and the pressure are established rigorously. In addition, a modified first-order projection scheme which leads to some improved error estimates is also proposed and analyzed. We also present the second-order projection method which is unconditionally stable. Ample numerical experiments are performed to confirm the theoretical predictions and demonstrate the efficiency of the methods.

참고문헌 (37)

  1. Ph. Angot, "Error analysis of the penalty-projection method for the time dependent Stokes equations", Int. J. Finite, 6(1), pp.26~ (2009)
  2. M. Case, "A connection between Scott-Vogelius elements and grad-div stabilization Taylor-Hood FE approximations of the Navier-Stokes equations", SIAM J. Numer. Anal., 49(4), pp.1461~1481 (2011)[DOI]
  3. J. Chen, "Constitutive equations of Micropolar electromagnetic fluids", J NonNewton Fluid., 166(14), pp.867~874 (2011)[DOI]
  4. A. J. Chorin, "Numerical solution of the Navier-Stokes equations", Math. Comp., 22, pp.745~762 (1968)[DOI]
  5. J. S. Dahler, "Angular momentum of continua", Nature, 192, pp.36~37 (1961)[DOI]
  6. J. S. Dahler, "Theory of structured continua. I. General consideration of angular momentum and polarization", Proc. R. Soc., 275(1363), pp.504~527 (1963)[DOI]
  7. B.-Q. Dong, "Global regularity of the 2D micropolar fluid flows with zero angular viscosity", J. Differential Equations, 249(1), pp.200~213 (2010)[DOI]
  8. A. C. Eringen, "Microcontinuum Field Theories. I. Foundations and Solids" (1999), Springer-Verlag
  9. A. C. Eringen, "Microcontinuum Field Theories. II. Fluent Media" (2001), Springer-Verlag
  10. G. P. Galdi, "A note on the existence and uniqueness of solutions of the micropolar fluid equations", Internat. J. Engrg. Sci., 15(2), pp.105~108 (1977)[DOI]
  11. V. Girault, "Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics, 5" (1986), Springer-Verlag
  12. J. L. Guermond, "An overview of projection methods for incompressible flows", Comput. Methods Appl. Mech. Engrg., 195(44-47), pp.6011~6045 (2006)[DOI]
  13. J. L. Guermond, "Velocity-correction projection methods for incompressible flows", SIAM J. Numer. Anal., 41(1), pp.112~134 (2003)[DOI]
  14. J. L. Guermond, "A new class of truly consistent splitting schemes for incompressible flows", J. Comput. Phys., 192(1), pp.262~276 (2003)[DOI]
  15. J. L. Guermond, "On the error estimates for the rotational pressure-correction projection methods", Math. Comp., 73(248), pp.1719~1737 (2004)
  16. F. Hecht, "New development in freefem++", J. Numer. Math., 20(3-4), pp.251~265 (2012)
  17. J. G. Heywood, "Finite-element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization", SIAM J. Numer. Anal., 27(2), pp.353~384 (1990)[DOI]
  18. Y. L. Jiang, "Semi-discrete Galerkin finite element method for the diffusive Peterlin viscoelastic model", Comput. Methods Appl. Math.[DOI]
  19. M. Jobelin, "A finite element penalty-projection method for incompressible flows", J. Comput. Phys., 217(2), pp.502~518 (2006)[DOI]
  20. A. Linke, "A connection between coupled and penalty projection timestepping schemes with FE spacial discretization for the Navier-Stokes equations", J. Numer. Math. (2016)[DOI]
  21. G. Lukaszewicz, "Micropolar Fluids, Modeling and Simulation in Science, Engineering and Technology" (1999), Birkhauser Boston, Inc
  22. E. Mallea-Zepeda, "A boundary control problem for micropolar fluids", J. Optim. Theory Appl., 169(2), pp.349~369 (2016)[DOI]
  23. M. Marion, "Navier-Stokes equations: theory and approximation, in Handbook of numerical analysis, Vol. VI, 503-688, Handb. Numer. Anal., VI" (1998)
  24. R. H. Nochetto, "The gauge-Uzawa finite element method. I. The Navier-Stokes equations", SIAM J. Numer. Anal., 43(3), pp.1043~1068 (2005)[DOI]
  25. R. H. Nochetto, "The micropolar Navier-Stokes equations: a priori error analysis", Math. Models Methods Appl. Sci., 24(7), pp.1237~1264 (2014)[DOI]
  26. R. H. Nochetto, "The equations of ferrohydrodynamics: modeling and numerical methods", Math. Models Methods Appl. Sci., 26(13), pp.2393~2449 (2016)[DOI]
  27. E. Ortega-Torres, "Optimal error estimate of the penalty finite element method for the micropolar fluid equations", Numer. Funct. Anal. Optim., 29(5-6), pp.612~637 (2008)[DOI]
  28. A. J. Salgado, "Convergence analysis of fractional time-stepping techniques for incompressible fluids with microstructure", J. Sci. Comput., 64(1), pp.216~233 (2015)[DOI]
  29. J. Shen, "On error estimates of projection methods for Navier-Stokes equations: firstorder schemes", SIAM J. Numer. Anal., 29(1), pp.57~77 (1992)[DOI]
  30. J. Shen, "Error estimates for finite element approximations of consistent splitting schemes for incompressible flows", Discrete Contin. Dyn. Syst. Ser. B, 8(3), pp.663~676 (2007)[DOI]
  31. R. Stavre, "The control of the pressure for a micropolar fluid", Z. Angew. Math. Phys., 53(6), pp.912~922 (2002)[DOI]
  32. R. Temam, "Une methode d'approximation de la solution des equations de Navier-Stokes", Bull. Soc. Math. France, 96, pp.115~152 (1968)
  33. R. Temam, "Navier-Stokes Equations" (1984), North-Holland Publishing Co
  34. J. van Kan, "A second-order accurate pressure-correction scheme for viscous incompressible flow", SIAM J. Sci. Statist. Comput., 7(3), pp.870~891 (1986)[DOI]
  35. Y. B. Yang, "Analysis of two decoupled time-stepping finite element methods for incompressible fluids with microstructure", Int. J. Comput. Math. (2017)[DOI]
  36. Y. B. Yang, "Numerical analysis and computation of a type of IMEX method for the timedependent natural convection problem", Comput. Methods Appl. Math., 16(2), pp.321~344 (2016)
  37. Y. B. Yang, "An explicitly uncoupled VMS stabilization finite element method for the time-dependent Darcy-Brinkman equations in double-diffusive convection", Numer. Algor.[DOI]

관련 논문AI추천

이 간행물의 다른 논문 (10건)