Analysis of Variable-Step/Non-autonomous Artificial Compression Methods

  • Robin Ming Chen
  • William LaytonEmail author
  • Michael McLaughlin


A standard artificial compression (AC) method for incompressible flow is
$$\begin{aligned}&\frac{u_{n+1}^{\varepsilon }-u_{n}^{\varepsilon }}{k}+u_{n+1}^{\varepsilon }\cdot \nabla u_{n+1}^{\varepsilon }+{\frac{1}{2}}u_{n+1}^{\varepsilon }\nabla \cdot u_{n+1}^{\varepsilon }+\nabla p_{n+1}^{\varepsilon }-\nu \Delta u_{n+1}^{\varepsilon }=f\text { ,} \\&\quad \varepsilon \frac{p_{n+1}^{\varepsilon }-p_{n}^{\varepsilon }}{k} +\nabla \cdot u_{n+1}^{\varepsilon }=0 \end{aligned}$$
for, typically, \(\varepsilon =k\) (timestep). It is fast, efficient and stable with accuracy \(O(\varepsilon +k)\). For adaptive (and thus variable) timestep \(k_{n}\) (and thus \(\varepsilon =\varepsilon _{n}\)) its long time stability is unknown. For variable \(k,\varepsilon \) this report shows how to adapt a standard AC method to recover a provably stable method. For the associated continuum AC model, we prove convergence of the \(\varepsilon =\varepsilon (t)\) artificial compression model to a weak solution of the incompressible Navier–Stokes equations as \(\varepsilon =\varepsilon (t)\rightarrow 0\). The analysis is based on space-time Strichartz estimates for a non-autonomous acoustic equation. Variable \(\varepsilon ,k\) numerical tests in 2d and 3d are given for the new AC method.


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Conflict of interest

The authors declare that they have no conflicts of interest.


  1. 1.
    Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E., Wells, G.N.: The FEniCS project version 1.5. Arch. Numer. Softw. 3, 9–23 (2015)Google Scholar
  2. 2.
    Aubin, J.: Un théorème de compacité. C. R. Acad. Sci. Paris 256, 5042–5044 (1963)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Berselli, L.C., Spirito, S.: On the construction of suitable weak solutions to the 3D Navier–Stokes equations in a bounded domain by an artificial compressibility method. Commun. Contemp. Math. 20, 1650064 (2018). MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762 (1968)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chorin, A.J.: On the convergence of discrete approximations to the Navier–Stokes equations. Math. Comput. 23, 341–353 (1969)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chaćon Rebollo, T., Lewandowski, R.: Mathematical and numerical foundations of turbulence models and applications. Modeling and Simulation in Science, Engineering and Technology. Springer, New York (2014)zbMATHGoogle Scholar
  7. 7.
    Charnyi, S., Heister, T., Olshanskii, M., Rebholz, L.: On conservation laws of Navier–Stokes Galerkin discretizations. J. Comput. Phys. 337, 289–308 (2017)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    DeCaria, V., Layton, W., McLaughlin, M.: A conservative, second order, unconditionally stable artificial compression method. CMAME 325, 733–747 (2017)ADSMathSciNetGoogle Scholar
  9. 9.
    Donatelli, D., Marcati, P.: A dispersive approach to the artificial compressibility approximations of the Navier–Stokes equations in 3D. J. Hyperbolic Differ. Equ. 3, 575–588 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Donatelli, D., Marcati, P.: Leray weak solutions of the incompressible Navier–Stokes system on exterior domains via the artificial compressibility method. Indiana Univ. Math. J. 59, 1831–1852 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Donatelli, D., Spirito, S.: Weak solutions of Navier–Stokes equations constructed by artificial compressibility method are suitable. J. Hyperbolic Differ. Equ. 8, 101–113 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ginibre, J., Velo, G.: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133, 50–68 (1995)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Guermond, J., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195, 6011–6045 (2006)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Guermond, J.-L., Minev, P.: High-order time stepping for the incompressible Navier–Stokes equations. SIAM J. Sci. Comput. 37–6, A2656–A2681 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Guermond, J.-L., Minev, P.: High-order time stepping for the Navier–Stokes equations with minimal computational complexity. JCAM 310, 92–103 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Guermond, J.-L., Minev, P.: High-order, adaptive time stepping scheme for the incompressible Navier–Stokes equations. Technical report 2018Google Scholar
  17. 17.
    Gunzburger, M.D.: Finite Element Methods for Viscous Incompressible Flows—A Guide to Theory, Practices, and Algorithms. Academic Press, Cambridge (1989)Google Scholar
  18. 18.
    Gresho, P.M., Sani, R.L.: Incompressible Flows and the Finite Element Method, vol. 2. Wiley, Chichester (2000)zbMATHGoogle Scholar
  19. 19.
    Johnston, H., Liu, J.-G.: Accurate, stable and efficient Navier–Stokes solvers based on an explicit treatment of the pressure term. JCP 199, 221–259 (2004)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955–980 (1998)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Klainerman, S., Macedon, M.: Space-time estimates for null forms and the local existence theorem. Commun. Pure Appl. Math. 46, 1221–1268 (1993)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kobel’kov, G.M.: Symmetric approximations of the Navier–Stokes equations. Sb. Math. 193, 1027–1047 (2002)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Layton, W.: An energy analysis of a degenerate hyperbolic partial differential equations. Apl. Mat. 29, 350–366 (1984)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Lions, J.L.: Sur l’existence de solutions des équations de Navier–Stokes. C. R. Acad. Sci. Paris 248, 2847–2849 (1959)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Lions, J.L.: Quelque méthodes de résolution des problemes aux limites non linéaires. Dunod-Gauth. Vill, Paris (1969)zbMATHGoogle Scholar
  26. 26.
    Lions, P.L.: Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models. Oxford University Press on Demand, Oxford (1996)Google Scholar
  27. 27.
    Mockenhaupt, G., Seeger, A., Sogge, C.D.: Local smoothing of Fourier integral operators and Carleson–Sjölin estimates. J. Am. Math. Soc. 6, 65–130 (1993)zbMATHGoogle Scholar
  28. 28.
    Ohwada, T., Asinari, P.: Artificial compressibility method revisited: asymptotic numerical method for incompressible Navier–Stokes equations. J. Comput. Phys. 229, 16981723 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Oskolkov, A.: On a quasi-linear parabolic system with a small parameter approximating the Navier–Stokes system. Zapiski Nauchnykh Seminarov POMI 21, 79–103 (1971)Google Scholar
  30. 30.
    Prohl, A.: Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier–Stokes Equations. Springer, Berlin (1997)CrossRefGoogle Scholar
  31. 31.
    Shen, J.: On error estimates of projection methods for the Navier–Stokes equations: first order schemes. SINUM 29, 57–77 (1992)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Shen, J.: On a new pseudocompressibility method for the incompressible Navier–Stokes equations. Appl. Numer. Math. 21, 71–90 (1996)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Smith, H.F.: A parametrix construction for wave equations with \(C^{1,1}\) coefficients. Ann. Inst. Fourier (Grenoble) 48, 797–835 (1998)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Söderlind, G., Fekete, I., Faragó, I.: On the 0-stability of multistep methods on smooth nonuniform grids, arXiv:1804.04553 (2018)
  35. 35.
    Sogge, C.: Lectures on Nonlinear Wave Equations. International Press, Cambridge (1995)zbMATHGoogle Scholar
  36. 36.
    Stein, E.M.: Harmonic Analysis (PMS-43), Volume 43: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (2016)Google Scholar
  37. 37.
    Strichartz, R.S.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44, 705–714 (1977)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Tataru, D.: Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation. Am. J. Math. 122, 349–376 (2000)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Tataru, D.: Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II. Am. J. Math. 123, 385–423 (2001)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Tataru, D.: Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III. J. Am. Math. Soc. 15, 419–442 (2002)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Temam, R.: Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires (I). Arch. Ration. Mech. Anal. 32, 135–153 (1969)CrossRefGoogle Scholar
  42. 42.
    Temam, R.: Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires (II). Arch. Ration. Mech. Anal. 33, 377–385 (1969)CrossRefGoogle Scholar
  43. 43.
    Temam, R.: Navier–Stokes equations and nonlinear functional analysis. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 41. SIAM, Philadelphia, PA (1983)Google Scholar
  44. 44.
    Temam, R.: Navier–Stokes equations. AMS Chelsea Publishing, Providence (2001)zbMATHGoogle Scholar
  45. 45.
    Van Kan, J.: A second order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Comput. 7, 870–891 (1986)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Yang, L., Badia, S., Codina, R.: A pseudo-compressible variational multiscale solver for turbulent incompressible flows. Comput. Mech. 58, 1051–1069 (2016)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Zeytounian, R.K.: Topics in Hyposonic Flow Theory, LN in Physics. Springer, Berlin (2006)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

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