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

## Abstract

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.

## Keywords

DAEs Difference-algebraic equations Navier-Stokes equations Strangeness index## **Mathematics Subject Classification (2010)**

65L80 65M12 35Q30 ## References

- 1.Altmann, R., Heiland, J.: Finite element decomposition and minimal extension for flow equations. ESAIM: Math. Model. Numer. Anal.
**49**(5):1489–1509 (2015)MathSciNetCrossRefGoogle Scholar - 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.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)MathSciNetCrossRefGoogle Scholar - 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.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.Benner, P., Heiland, J.: Time-dependent Dirichlet conditions in finite element discretizations. ScienceOpen Research, 1–18 (2015)Google Scholar
- 7.Braack, M., Mucha, P.B.: Directional do-nothing condition for the Navier-Stokes equations. J. Comput. Math.
**32**(5), 507–521 (2014)MathSciNetCrossRefGoogle Scholar - 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)MathSciNetCrossRefGoogle Scholar - 9.Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)CrossRefGoogle Scholar
- 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)MathSciNetCrossRefGoogle Scholar - 11.Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comput.
**22**, 745–762 (1968)MathSciNetCrossRefGoogle Scholar - 12.Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics, 3rd edn. Springer, New York (1993)CrossRefGoogle Scholar
- 13.Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
- 14.Dai, L.: Singular Control Systems. Lecture Notes in Control and Information Sciences, vol. 118. Springer, Berlin (1989)Google Scholar
- 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.Emmrich, E., Mehrmann, V.: Operator differential-algebraic equations arising in fluid dynamics. Comp. Methods Appl. Math.
**13**(4), 443–470 (2013)MathSciNetzbMATHGoogle Scholar - 17.Feireisl, E., Karper, T.G., Pokorný, M.: Mathematical Theory of Compressible Viscous Fluids. Analysis and Numerics. Birkhäuser/Springer, Basel (2016)Google Scholar
- 18.Ferziger, J.H., Perić, M.: Computational Methods for Fluid Dynamics, 3rd edn. Springer, Berlin (2002)CrossRefGoogle Scholar
- 19.Fujita, H., Kato, T.: On the Navier-Stokes initial value problem. I. Arch. Ration. Mech. Anal.
**16**, 269–315 (1964)MathSciNetCrossRefGoogle Scholar - 20.Gaul, A.: Krypy – a Python toolbox of iterative solvers for linear systems, commit: 36e40e1d (2017). https://github.com/andrenarchy/krypy
- 21.Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer, Berlin (1986)Google Scholar
- 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.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.Gresho, P.M., Sani, R.L.: Incompressible Flow and the Finite Element Method. Vol. 2: Isothermal Laminar Flow. Wiley, Chichester (2000)Google Scholar
- 25.Griebel, M., Dornseifer, T., Neunhoeffer, T.: Numerical Simulation in Fluid Dynamics. A Practical Introduction. SIAM, Philadelphia (1997)Google Scholar
- 26.Ha, P.: Analysis and numerical solutions of delay differential algebraic equations. Ph.D. thesis, Technische Universität Berlin (2015)Google Scholar
- 27.Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)CrossRefGoogle Scholar
- 28.Hairer, E., Lubich, C., Roche, M.: The numerical solution of differential-algebraic systems by Runge-Kutta methods. Springer, Berlin (1989)CrossRefGoogle Scholar
- 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.He, X., Vuik, C.: Comparison of some preconditioners for the incompressible Navier-Stokes equations. Numer Math. Theory Methods Appl
**9**(2), 239–261 (2016)MathSciNetCrossRefGoogle Scholar - 31.Heiland, J.: Decoupling and optimization of differential-algebraic equations with application in flow control. Ph.D. thesis, TU Berlin (2014). http://opus4.kobv.de/opus4-tuberlin/frontdoor/index/index/docId/5243
- 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.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.Hinze, M.: Optimal and instantaneous control of the instationary Navier-Stokes equations. Habilitationsschrift, Institut für Mathematik, Technische Universität Berlin (2000)Google Scholar
- 35.Karniadakis, G., Beskok, A., Narayan, A.: Microflows and Nanoflows. Fundamentals and Simulation. Springer, New York (2005)Google Scholar
- 36.Kunkel, P., Mehrmann, V.: Canonical forms for linear differential-algebraic equations with variable coefficients. J. Comput. Appl. Math.
**56**(3), 225–251 (1994)MathSciNetCrossRefGoogle Scholar - 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)MathSciNetCrossRefGoogle Scholar - 38.Kunkel, P., Mehrmann, V.: Index reduction for differential-algebraic equations by minimal extension. Z. Angew. Math. Mech.
**84**(9), 579–597 (2004)MathSciNetCrossRefGoogle Scholar - 39.Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution. European Mathematical Society Publishing House, Zürich (2006)CrossRefGoogle Scholar
- 40.Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach Science Publishers, New York (1969)zbMATHGoogle Scholar
- 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.Layton, W.: Introduction to the Numerical Analysis of Incompressible Viscous Flows. SIAM, Philadelphia (2008)CrossRefGoogle Scholar
- 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.LeVeque, R.J.: Numerical Methods for Conservation Laws, 2nd edn. Birkhäuser, Basel (1992)Google Scholar
- 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.Mattsson, S.E., Söderlind, G.: Index reduction in differential-algebraic equations using dummy derivatives. SIAM J. Sci. Comput.
**14**(3), 677–692 (1993)MathSciNetCrossRefGoogle Scholar - 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)MathSciNetCrossRefGoogle Scholar - 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)MathSciNetCrossRefGoogle Scholar - 49.Pironneau, O.: Finite Element Methods for Fluids. Wiley/Masson, Chichester/Paris (1989) Translated from the FrenchGoogle Scholar
- 50.Raymond, J.-P.: Feedback stabilization of a fluid-structure model. SIAM J. Cont. Optim.
**48**(8), 5398–5443 (2010)MathSciNetCrossRefGoogle Scholar - 51.Reis, T.: Systems Theoretic Aspects of PDAEs and Applications to Electrical Circuits. Shaker, Aachen (2006)Google Scholar
- 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.Roubí\({\check {\text c}}\)ek, T.: Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel (2005)Google Scholar
- 54.Saint-Raymond, L.: Hydrodynamic Limits of the Boltzmann Equation. Springer, Berlin (2009)CrossRefGoogle Scholar
- 55.Shen, J.: On error estimates of the penalty method for unsteady Navier-Stokes equations. SIAM J. Numer. Anal.
**32**(2), 386–403 (1995)MathSciNetCrossRefGoogle Scholar - 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.Tartar, L.: An Introduction to Navier–Stokes Equation and Oceanography. Springer, New York (2006)CrossRefGoogle Scholar
- 58.Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam (1977)Google Scholar
- 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)MathSciNetCrossRefGoogle Scholar - 60.Turek, S.: Efficient Solvers for Incompressible Flow Problems. An Algorithmic and Computational Approach. Springer, Berlin (1999)Google Scholar
- 61.Weickert, J.: Navier-Stokes equations as a differential-algebraic system. Preprint SFB393/96-08, Technische Universität Chemnitz-Zwickau (1996)Google Scholar
- 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.Williamson, C.H.K.: Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech.
**28**(1), 477–539 (1996)MathSciNetCrossRefGoogle Scholar - 64.Zeidler, E.: Nonlinear Functional Analysis and its Applications. II/A: Linear Monotone Operators. Springer, Berlin (1990)Google Scholar