Continuous, Semi-discrete, and Fully Discretised Navier-Stokes Equations

  • R. Altmann
  • J. HeilandEmail author
Part of the Differential-Algebraic Equations Forum book series (DAEF)


The Navier-Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretisation. We analyse the semi-discrete equations – a semi-explicit nonlinear DAE – in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analysing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier-Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.


DAEs Difference-algebraic equations Navier-Stokes equations Strangeness index 

Mathematics Subject Classification (2010)

65L80 65M12 35Q30 


  1. 1.
    Altmann, R., Heiland, J.: Finite element decomposition and minimal extension for flow equations. ESAIM: Math. Model. Numer. Anal. 49(5):1489–1509 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Altmann, R., Heiland, J.: Regularization and Rothe discretization of semi-explicit operator DAEs. Int. J. Numer. Anal. Model. 15(3), 452–477 (2018)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Arnold, M., Strehmel, K., Weiner, R.: Half-explicit Runge-Kutta methods for semi-explicit differential-algebraic equations of index 1. Numer. Math. 64(1), 409–431 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Behr, M., Benner, P., Heiland, J.: Example setups of Navier-Stokes equations with control and observation: spatial discretization and representation via linear-quadratic matrix coefficients. Technical Report (2017). arXiv:1707.08711Google Scholar
  5. 5.
    Benner, P., Heiland, J.: LQG-balanced truncation low-order controller for stabilization of laminar flows. In: King, R. (ed.) Active Flow and Combustion Control 2014, pp. 365–379. Springer, Berlin (2015)Google Scholar
  6. 6.
    Benner, P., Heiland, J.: Time-dependent Dirichlet conditions in finite element discretizations. ScienceOpen Research, 1–18 (2015)Google Scholar
  7. 7.
    Braack, M., Mucha, P.B.: Directional do-nothing condition for the Navier-Stokes equations. J. Comput. Math. 32(5), 507–521 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Brasey, V., Hairer, E.: Half-explicit Runge–Kutta methods for differential-algebraic systems of index 2. SIAM J. Numer. Anal. 30(2), 538–552 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)zbMATHCrossRefGoogle Scholar
  10. 10.
    Campbell, S.: A general form for solvable linear time varying singular systems of differential equations. SIAM J. Math. Anal. 18(4), 1101–1115 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comput. 22, 745–762 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics, 3rd edn. Springer, New York (1993)zbMATHCrossRefGoogle Scholar
  13. 13.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  14. 14.
    Dai, L.: Singular Control Systems. Lecture Notes in Control and Information Sciences, vol. 118. Springer, Berlin (1989)Google Scholar
  15. 15.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press, Oxford (2005)zbMATHGoogle Scholar
  16. 16.
    Emmrich, E., Mehrmann, V.: Operator differential-algebraic equations arising in fluid dynamics. Comp. Methods Appl. Math. 13(4), 443–470 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Feireisl, E., Karper, T.G., Pokorný, M.: Mathematical Theory of Compressible Viscous Fluids. Analysis and Numerics. Birkhäuser/Springer, Basel (2016)Google Scholar
  18. 18.
    Ferziger, J.H., Perić, M.: Computational Methods for Fluid Dynamics, 3rd edn. Springer, Berlin (2002)zbMATHCrossRefGoogle Scholar
  19. 19.
    Fujita, H., Kato, T.: On the Navier-Stokes initial value problem. I. Arch. Ration. Mech. Anal. 16, 269–315 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gaul, A.: Krypy – a Python toolbox of iterative solvers for linear systems, commit: 36e40e1d (2017).
  21. 21.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer, Berlin (1986)Google Scholar
  22. 22.
    Glowinski, R.: Finite element methods for incompressible viscous flow. In: Numerical Methods for Fluids (Part 3). Handbook of Numerical Analysis, vol. 9, pp. 3–1176. Elsevier, Burlington (2003)Google Scholar
  23. 23.
    Gresho, P.M.: On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. I: Theory. Int. J. Numer. Methods Fluids 11(5), 587–620 (1990)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Gresho, P.M., Sani, R.L.: Incompressible Flow and the Finite Element Method. Vol. 2: Isothermal Laminar Flow. Wiley, Chichester (2000)Google Scholar
  25. 25.
    Griebel, M., Dornseifer, T., Neunhoeffer, T.: Numerical Simulation in Fluid Dynamics. A Practical Introduction. SIAM, Philadelphia (1997)Google Scholar
  26. 26.
    Ha, P.: Analysis and numerical solutions of delay differential algebraic equations. Ph.D. thesis, Technische Universität Berlin (2015)Google Scholar
  27. 27.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)zbMATHCrossRefGoogle Scholar
  28. 28.
    Hairer, E., Lubich, C., Roche, M.: The numerical solution of differential-algebraic systems by Runge-Kutta methods. Springer, Berlin (1989)zbMATHCrossRefGoogle Scholar
  29. 29.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer Series in Computational Mathematics. Springer, Berlin (2006)Google Scholar
  30. 30.
    He, X., Vuik, C.: Comparison of some preconditioners for the incompressible Navier-Stokes equations. Numer Math. Theory Methods Appl 9(2), 239–261 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Heiland, J.: Decoupling and optimization of differential-algebraic equations with application in flow control. Ph.D. thesis, TU Berlin (2014).
  32. 32.
    Heinrich, J.C., Vionnet, C.A.: The penalty method for the Navier-Stokes equations. Arch. Comput. Method E 2, 51–65 (1995)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Heywood, J.G., Rannacher, R.: Finite-element approximation of the nonstationary Navier–Stokes problem. IV: Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27(2), 353–384 (1990)zbMATHGoogle Scholar
  34. 34.
    Hinze, M.: Optimal and instantaneous control of the instationary Navier-Stokes equations. Habilitationsschrift, Institut für Mathematik, Technische Universität Berlin (2000)Google Scholar
  35. 35.
    Karniadakis, G., Beskok, A., Narayan, A.: Microflows and Nanoflows. Fundamentals and Simulation. Springer, New York (2005)Google Scholar
  36. 36.
    Kunkel, P., Mehrmann, V.: Canonical forms for linear differential-algebraic equations with variable coefficients. J. Comput. Appl. Math. 56(3), 225–251 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Kunkel, P., Mehrmann, V.: Analysis of over- and underdetermined nonlinear differential-algebraic systems with application to nonlinear control problems. Math. Control Signals Syst. 14(3), 233–256 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Kunkel, P., Mehrmann, V.: Index reduction for differential-algebraic equations by minimal extension. Z. Angew. Math. Mech. 84(9), 579–597 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution. European Mathematical Society Publishing House, Zürich (2006)zbMATHCrossRefGoogle Scholar
  40. 40.
    Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach Science Publishers, New York (1969)zbMATHGoogle Scholar
  41. 41.
    Landau, L.D., Lifshits, E.M.: Fluid Mechanics. Course of Theoretical Physics, vol. 6, 2nd edn. Elsevier, Amsterdam (1987). Transl. from the Russian by J. B. Sykes and W. H. Reid.Google Scholar
  42. 42.
    Layton, W.: Introduction to the Numerical Analysis of Incompressible Viscous Flows. SIAM, Philadelphia (2008)zbMATHCrossRefGoogle Scholar
  43. 43.
    Leray, J.: étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)Google Scholar
  44. 44.
    LeVeque, R.J.: Numerical Methods for Conservation Laws, 2nd edn. Birkhäuser, Basel (1992)Google Scholar
  45. 45.
    Logg, A., Ølgaard, K.B., Rognes, M.E., Wells, G.N.: FFC: the FEniCS form compiler. In: Automated Solution of Differential Equations by the Finite Element Method, pp. 227–238. Springer, Berlin (2012)Google Scholar
  46. 46.
    Mattsson, S.E., Söderlind, G.: Index reduction in differential-algebraic equations using dummy derivatives. SIAM J. Sci. Comput. 14(3), 677–692 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Noack, B.R., Afanasiev, K., Morzyński, M., Tadmor, G., Thiele, F.: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Nguyen, P.A., Raymond, J.-P.: Boundary stabilization of the Navier–Stokes equations in the case of mixed boundary conditions. SIAM J. Control. Optim. 53(5), 3006–3039 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Pironneau, O.: Finite Element Methods for Fluids. Wiley/Masson, Chichester/Paris (1989) Translated from the FrenchGoogle Scholar
  50. 50.
    Raymond, J.-P.: Feedback stabilization of a fluid-structure model. SIAM J. Cont. Optim. 48(8), 5398–5443 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Reis, T.: Systems Theoretic Aspects of PDAEs and Applications to Electrical Circuits. Shaker, Aachen (2006)Google Scholar
  52. 52.
    Reynolds, O.: Papers on Mechanical und Physical Subjects. Volume III. The Sub-mechanics of the Universe. Cambridge University Press, Cambridge (1903)Google Scholar
  53. 53.
    Roubí\({\check {\text c}}\)ek, T.: Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel (2005)Google Scholar
  54. 54.
    Saint-Raymond, L.: Hydrodynamic Limits of the Boltzmann Equation. Springer, Berlin (2009)zbMATHCrossRefGoogle Scholar
  55. 55.
    Shen, J.: On error estimates of the penalty method for unsteady Navier-Stokes equations. SIAM J. Numer. Anal. 32(2), 386–403 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Steinbrecher, A.: Numerical Solution of Quasi-Linear Differential-Algebraic Equations and Industrial Simulation of Multibody Systems. Ph.D. thesis, Technische Universität Berlin (2006)Google Scholar
  57. 57.
    Tartar, L.: An Introduction to Navier–Stokes Equation and Oceanography. Springer, New York (2006)zbMATHCrossRefGoogle Scholar
  58. 58.
    Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam (1977)Google Scholar
  59. 59.
    Taylor, C., Hood, P.: A numerical solution of the Navier-Stokes equations using the finite element technique. Int. J. Comput. Fluids 1(1), 73–100 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Turek, S.: Efficient Solvers for Incompressible Flow Problems. An Algorithmic and Computational Approach. Springer, Berlin (1999)Google Scholar
  61. 61.
    Weickert, J.: Navier-Stokes equations as a differential-algebraic system. Preprint SFB393/96-08, Technische Universität Chemnitz-Zwickau (1996)Google Scholar
  62. 62.
    Weickert, J.: Applications of the theory of differential-algebraic equations to partial differential equations of fluid dynamics. Ph.D. thesis, Fakultät für Mathematik, Technische Universität Chemnitz (1997)Google Scholar
  63. 63.
    Williamson, C.H.K.: Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28(1), 477–539 (1996)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications. II/A: Linear Monotone Operators. Springer, Berlin (1990)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AugsburgAugsburgGermany
  2. 2.Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany

Personalised recommendations