Spatial and Temporal Adaptivity in Numerical Studies of Instabilities, with Applications to Fluid Flows

Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 50)


In this article we discuss how to formulate numerical methods for calculating branches of steady solutions, periodic orbits and bifurcations of partial differential equations that are adaptive in both space and time. The methods are implemented within the open-source software framework oomph-lib and examples of their use in current research problems of fluid flow past cylinders and free surface flows on rotating cylinders are presented.


Adaptive Temperature Flow Past Spatial Adaptivity Augmented System Adaptive Space-time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was initiated in 2012 while the author was a visiting scientist at Sandia National Laboratories, hosted by A.G. Salinger. The ideas presented within have been refined over numerous discussions with many colleagues at workshops and conferences and many of the computations have been conducted by Ph.D. students. Here, I would particularly like to thank A. von Borries Lopes who worked on the rotating cylinder problem and P. Matharu who is working on the flow past a cylinder. Oomph-lib would not exist without the tireless work of my colleague M. Heil and much of the bifurcation tracking and detection capabilities were developed from extensive discussions with R. Hewitt, T. Mullin and K.A. Cliffe. U. Thiele contributed to the work on partially wetting fluids in Sect. 5.1. Finally, I would also like to thank L. van Veen and S. Altmeyer for their comments which have helped to improve an earlier draft of this chapter.


  1. 1.
    Amestoy, P.R., Duff, I.S., Koster, J., L’Excellent, J.Y.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23(1), 15–41 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Amestoy, P.R., Guermouche, A., L’Excellent, J.Y., Pralet, S.: Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32(2), 136–156 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aruliah, D.A., van Veen, L., Dubitski, A.: A parallel adaptive method for pseudo-arclength continuation. J. Phys.: Conf. Ser. 385(1), 012–008 (2012). Scholar
  4. 4.
    Balay S, Abhyankar S, Adams MF, Brown J, Brune P, Buschelman K, Dalcin L, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Rupp K, Smith BF, Zampini S, Zhang H, Zhang H (2016) PETSc Web page.
  5. 5.
    Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Springer, Berlin (2003)CrossRefGoogle Scholar
  6. 6.
    Cairncross, R.A., Schunk, P.R., Baer, T.A., Rao, R.R., Sackinger, P.A.: A finite element method for free surface flows of incompressible fluids in three dimensions. part i. boundary fitted mesh motion. Int. J. Numer. Methods Fluids 33, 375–403 (2000)CrossRefGoogle Scholar
  7. 7.
    Cliffe, K.A.: ENTWIFE (release 6.3) reference manual. Oxford, UK: Harwell Laboratory (1996).
  8. 8.
    Cliffe, K.A., Tavener, S.J.: The effect of cylinder rotation and blockage ratio on the onset of periodic flows. J. Fluid Mech. 501, 125133 (2004). Scholar
  9. 9.
    Cliffe, K.A., Spence, A., Tavener, S.J.: The numerical analysis of bifurcation problems with application to fluid mechanics. Acta Numerica 9, 39–131 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cliffe, K.A., Hall, E.J.C., Houston, P., Phipps, E.T., Salinger, A.G.: Adaptivity and a posteriori error control for bifurcation problems iii: Incompressible fluid flow in open systems with z 2 symmetry. J. Sci. Comput. 47(3), 389–418 (2011). Scholar
  11. 11.
    Cliffe, K.A., Hall, E.J.C., Houston, P., Phipps, E.T., Salinger, A.G.: Adaptivity and a posteriori error control for bifurcation problems iii: Incompressible fluid flow in open systems with o(2) symmetry. J. Sci. Comput. 52(1), 153–179 (2012). Scholar
  12. 12.
    Cliffe, K.A., Hall, E.J.C., Houston, P.: Application of hp–adaptive discontinuous galerkin methods to bifurcation phenomena in pipe flows. In: Cangiani, A., Davidchack, R.L., Georgoulis, E., Gorban, A.N., Levesley, J., Tretyakov, M.V. (eds) Numerical Mathematics and Advanced Applications 2011: Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications, Leicester, September 2011, Springer, Berlin, Heidelberg, pp. 333–340 (2013)Google Scholar
  13. 13.
    Doedel, E., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X.: Auto97: Continuation and bifurcation software for ordinary differential equations (with homcont). Concordia University, Technical Report (1997)Google Scholar
  14. 14.
    Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers. Oxford University Press, Oxford (2005)zbMATHGoogle Scholar
  15. 15.
    Geuzaine, C., Remacle, J.F.: Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009). Scholar
  16. 16.
    Gibson JF (2014) Channelflow: A spectral Navier-Stokes simulator in C++. Technical Report, U. New Hampshire.
  17. 17.
    Gollub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)Google Scholar
  18. 18.
    Govaerts, W.J.F.: Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  19. 19.
    Gresho PM, Sani RL (1998) Incompressible flow and the finite element method. Volume 1: Advection-diffusion and isothermal laminar flow. Wiley, Inc., New York, NY (United States)Google Scholar
  20. 20.
    Heil, M., Hazel, A.L.: oomph-lib - an object-oriented multi-physics finite-element library. In: Bungartz, H.J., Schäfer, M. (eds.) Fluid-Structure Interaction: Modelling, pp. 19–49. Springer, Simulation, Optimisation (2006)Google Scholar
  21. 21.
    Heil, M., Rosso, J., Hazel, A.L., Brns, M.: Topological fluid mechanics of the formation of the krmn-vortex street. J. Fluid Mech. 812, 199221 (2017). Scholar
  22. 22.
    Hernandez, V., Roman, J.E., Vidal, V.: SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans Math Softw. 31(3), 351–362 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Heroux, M.A., Bartlett, R.A., Howle, V.E., Hoekstra, R.J., Hu, J.J., Kolda, T.G., Lehoucq, R.B., Long, K.R., Pawlowski, R.P., Phipps, E.T., Salinger, A.G., Thornquist, H.K., Tuminaro, R.S., Willenbring, J.M., Williams, A., Stanley, K.S.: An overview of the trilinos project. ACM Trans. Math. Softw. 31(3):397–423 (2005). Scholar
  24. 24.
    Hewitt, R.E., Hazel, A.L., Clarke, R.J., Denier, J.P.: Unsteady flow in a rotating torus after a sudden change in rotation rate. J. Fluid Mech. 688, 88119 (2011). Scholar
  25. 25.
    Hydon, P.E.: Symmetry Method for Differential Equations. Cambridge University Press, A Beginner’s Guide (2000)Google Scholar
  26. 26.
    Karniadakis, G.E.M., Sherwin, S.: Spectral/\(hp\) Element Methods for Computational Fluid Dynamics. Numerical Methods and Scientific Computation, 2nd edn. Oxford University Press, Oxford (2005)CrossRefGoogle Scholar
  27. 27.
    Kawahara, G., Uhlmann, M., van Veen, L.: The significance of simple invariant solutions in turbulent flows. Ann. Rev. Fluid Mech. 44(1), 203–225 (2012). Scholar
  28. 28.
    Keller, H.E.: Numerical solution of bifurcation and non-linear eigenvalue problems. In: Rabinowitz, P. (ed.) Applications of Bifurcation Theory. Academic Press, New York (1977)Google Scholar
  29. 29.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Spinger, Berlin (1998)zbMATHGoogle Scholar
  30. 30.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, vol. 112, 3rd edn. Springer, New York (2010)Google Scholar
  31. 31.
    Li, X.S.: An overview of SuperLU: algorithms, implementation, and user interface. ACM Trans. Math. Softw. 31(3), 302–325 (2005)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Lopes, Av.B. (2018) Dynamics of free surface flows on rotating cylinders. Ph.D. thesis, The University of ManchesterGoogle Scholar
  33. 33.
    AvB, Lopes, Thiele, U., Hazel, A.L.: On the multiple solutions of coating and rimming flows on rotating cylinders. J. Fluid Mech. 835, 540574 (2018a). Scholar
  34. 34.
    AvB, Lopes, Thiele, U., Hazel, A.L.: On the multiple solutions of coating and rimming flows on rotating cylinders. J. Fluid Mech. 835, 540574 (2018b). Scholar
  35. 35.
    Lust, K., Roose, D., Spence, A., Champneys, A.R.: An adaptive newton-picard algorithm with subspace iteration for computing periodic solutions. SIAM J. Sci. Comput. 19(4), 1188–1209 (1998)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Moffatt, H.K.: Behaviour of a viscous film on the outer surface of a rotating cylinder. Journal de Mécanique 16 (1977)Google Scholar
  37. 37.
    Net, M., Snchez, J.: Continuation of bifurcations of periodic orbits for large-scale systems. SIAM J. Appl. Dyn. Syst. 14(2), 674–698 (2015). Scholar
  38. 38.
    Ruschak, K.J.: A method for incorporating free boundaries with surface tension in finite element fluid-flow simulators. Int. J. Numer. Methods Eng. 15, 639–648 (1980)CrossRefGoogle Scholar
  39. 39.
    Salinger, A., Romero, L., Pawlowski, R., Wilkes, E., Lehoucq, R., Burroughs, B., Bou-Rabee, N.: Loca: Library of continuation algorithms (2001).
  40. 40.
    Sanchez, J., Net, M.: Numerical continuation methods for large-scale dissipative dynamical systems. Eur. Phys. J. Spec. Top 225:2465–2486.
  41. 41.
    Schöberl, J.: Netgen an advancing front 2d/3d-mesh generator based on abstract rules. Comput. Vis. Sci. 1(1), 41–52 (1997). Scholar
  42. 42.
    Shewchuk, J.R.: Triangle: Engineering a 2D quality mesh generator and delaunay triangulator. In: Lin, M.C. (ed.) Computational, Applied, Geometry, Towards Geometric Engineering, pp. 203–222. Springer (1996)CrossRefGoogle Scholar
  43. 43.
    Si, H.: Tetgen, a delaunay-based quality tetrahedral mesh generator. ACM Trans. Math. Softw. 41(2):11:1–11:36 (2015). Scholar
  44. 44.
    Tavener, S.J.: Symmetric and nonsymmetric equilibria of a rod-and-spring model. IMA J. Appl. Math. 49(1), 73–102 (1992)MathSciNetCrossRefGoogle Scholar
  45. 45.
    The CGAL Project CGAL User and Reference Manual, 4.11.1 edn. CGAL Editorial Board (2018).
  46. 46.
    Thiele, U.: On the depinning of a drop of partially wetting liquid on a rotating cylinder. J. Fluid Mech. 671, 121–136 (2011)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Uecker, H., Wetzel, D., Rademacher, J.D.M.: pde2path - a matlab package for continuation and bifurcation in 2d elliptic systems. Numer. Math.: Theory, Methods Appl. 7(1), 58106 (2014). Scholar
  48. 48.
    Walker, H.F.: An adaptation of krylov subspace methods to path following problems. SIAM J. Sci. Comput. 21(3), 1191–1198 (1999). Scholar
  49. 49.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, Berlin (1990)CrossRefGoogle Scholar
  50. 50.
  51. 51.
    Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. part 1: The recovery technique. Int. J. Numer. Methods Eng. 33, 1331–1364 (1992)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsThe University of ManchesterManchesterUK

Personalised recommendations